Download Unit 1: Probability

Unit 1: Probability
Sets and Venn Diagrams
A.1
Objectives:
Define principles of inclusion and exclusion when working with sets and/or events. Introduce
the concepts of Union of sets (A » B) and intersection of sets (A « B).
Notes:
A set is a collection of items. Each item within the set is called a member of that set.
We can designate a set in many ways;
- by listing the members: A  {0, 1, 2, 3, 4, }
an ellipsis(…) means the set continues without end.
- by defining the set B  {x | x  I , x is odd }
B is the set of all integers x., such that x is odd.
"  "means ‘is a member of’
"|" -means ‘such that’
- by using a Venn diagram;
R
S
Q
The set Q contains the sets R & S.
The empty set is a set that has no members. It can be written {} or Ø but never {Ø}
The Union() of two sets includes all of the members in both sets.
- If A = {1, 2, 3, 4} and B = {3, 4, 5, 6} then A  B = {1, 2, 3, 4, 5, 6}
or in the diagram below, the shaded portion is R  S
R
S
Q
The Intersection() of two sets is all of the members that are common to both sets.
- If A = {1, 2, 3, 4} and B = {3, 4, 5, 6} then A  B = {3, 4}
or in the diagram below, the shaded portion is R  S
R
S
Q
If two sets have no members in common, the intersection is the empty set {} or Ø.
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Unit 1: Probability
Examples:
Given the sets; P = {-6, -4, -2, 0, 2, 4, 6}, Q = {-9, -6, -3, 0, 3, 6, 9}
and R = { -3, -1, 1, 3};
Find:
a. P  R
solution: P  R = {-6, -4, -3, -2, -1, 0, 1, 3, 4, 6} Union () means all of both.
b. Q  R
solution: Q  R = {-3, 3} Intersection () means all elements common to both.
c. P  R  Q
solution: P  R  Q = {-6, -4, -3, -2, -1, 0, 1, 3, 4, 6}  Q that is {P  R}  Q
= {-6, -3, 0, 3, 6}
d. P  R
solution: P  R = Ø or {} The sets have nothing in common.
Practice Questions 1: Union and Intersection of sets
1. For the following questions write true or false:
a.{5, 7, 9, 10}  {7, 9, 11, 15} = {7, 9}
b.{2, 1, 7}  {1, 5, 9} = {1}
c.{3, 2, 5, 9}  {2, 7, 8, 10} = {2, 3, 5, 7, 8, 9, 10}
d.{1, 2, 4}  { 1, 2, 4} = {1, 2, 4}
2. Given the following sets:
W = {2, 3, 4, 5}, X = {3, 5, 7, 9}, Y = {2, 4, 5, 7, 9}, and Z = {2, 3, 4, 5, 7, 9}
List the elements in each of the following;
a. W  X
b. X  Y
c. W  Z
d. W  (X  Y)
3. Use the diagram at right to find;
Q
11
3
a. A  B  C  D
C
B
b. A  B  C  D
10
A
12
5
15
4
c. (A  B)  (C  D)
D
14
2
1
3
8
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18
Unit 1: Probability
4. A survey of 80 first year students at the SIFC showed that:
36 take english
32 take history
32 take science
16 take history and english
16 take science and history
14 take science and english
6 take all three
Draw a Venn diagram to answer the following questions.
a. How many students take English and neither of the others.
b. How many students take none of the three courses
c. How many students take history and neither of the others.
d. How many students take science and history but not english.
e.How many students do not take science.
5. In each of the following copy the diagram and shade the indicated region.
a. R  S
Q
b. R  S
Q
S
T
R
c. R  T
Q
T
R
b. R  T
Q
S
R
S
T
S
R
T
6. Draw a Venn diagram to illustrate the following situation and then answer the questions.
From the grade 12 student request forms, 163 students want math, 95 students want
physics, 124 want chemistry and 17 want none of these. In the group, 61 want both math and
physics, 67 want both math and chemistry, 49 want both chemistry and physics and 27 want
all three.
a. How many students wanted only math?
b. How many students wanted only physics?
c. How many students wanted only chemistry?
d. How many request forms were there?
7. If A  B contains 17 elements, while A has 46 elements and B has 39 elements, how many
elements are in A  B?
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Unit 1: Probability
Probability, Mutually Exclusive, Dependent and Independent Events
Objectives:
A.2
favorable outcomes
event space

total possible outcomes sample space
To determine the probability of mutually exclusive events.
A.3
To determine the probability of two or more independent events.
A.4
To determine the probability of dependent events.
A.5
To set up, analyze, estimate, and solve word problems based on objectives 1-4.
Calculate probabilities using the formula; P ( a ) 
Notes:
In any experiment, the set of all possible outcomes is the sample space for the experiment.
Any subset of possible outcomes is called an event.
For example: When a coin is tossed three times, the sample space would be;
{(H, H, H) (H, H, T) (H, T, H) (H, T, T) (T, H, H) (T, H, T) (T, T, H) (T, T, T)}
{(H, H, H) (T, T, T)} is the event that all three tosses are the same.
List the event that two of the tosses are the same.
Solution {(H, H, T) (H, T, H) (H, T, T) (T, H, T) (T, H, H) (T, T, H)}
Probability of an event can be calculated using the formula;
favorable outcomes
event space
P(a) 

total possible outcomes sample space
In the example above we can calculate the probability that three tosses of a coin will show
the same face;
event space
2 1
P(a) 
 
sample space 8 4
In a situation where there are n equally likely outcomes and e is a simple event (one of the
1
outcomes) then; P ( e ) 
n
1
For example: In a coin toss there are two possible outcomes.  P ( H ) 
2
Generally the probability of an event that is certain to happen is 1.0.
6
For example the probability of rolling a 6 or less on a single die: P (6 or less )   1.0
The
6
probability of something that will never happen is 0.
0
For example the probability of rolling a 7 on a single die: P (7)   0
6
The probability of the converse(opposite) of an event is = 1 - the probability of the event.
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Unit 1: Probability
For example the probability that three tosses of a coin will not show the same face is;
1 3
P ( same )  1  P ( same )  1  
Note that P ( same ) means P ( not same )
4 4
Probability of Related Events:
Consider the act of rolling two six-sided dice. The sample space would be;
,,
16
15
,,

14
,,

,,
13
12
,,

,,
11
66 
25 35
,, 45 55
,, 65 

24 34
,, 44 54
,, 64 

2,,33 34 ,,35 36 , 3
22 32
,, 42 52
,, 62 

21 31
,, 41 51
,, 61 
26
36
,,
46
56
,,
It is possible to specify many different events; for example the event that one of the dice
shows a 3;
,,
16
15
,,

14
,,

,,
13
12
,,

,,
11
66 
25 35
,, 45 55
,, 65 

24 34
,, 44 54
,, 64 

2,,33 34 ,,35 36 , 3
22 32
,, 42 52
,, 62 

21 31
,, 41 51
,, 61 
26
36
,,
46
56
,,
10 5

we do not count 3,3 because both dice show a 3.
36 18
or the event that the total of the dice is 8;
P (3) 
,,
16
15
 ,,
14
,,

,,
13
12
,,

11
 ,,
66 
25 35
,, 45 55
,, 65 
24 34
,, 44 54
,, 64 

2,,33 34 ,,35 36 , 3
22 32
,, 42 52
,, 62 

21 31
,, 41 51
,, 61 
26
36
,,
46
P (8) 
56
,,
5
36
We can combine these events in two ways;
1. The event that one of the dice shows a three and the sum is 8;
,,
16
15
,,

14
,,

13
 ,,
12
,,

,,
11
P (3  8) 
66 
25 35
,, 45 55
,, 65 

24 34
,, 44 54
,, 64 

2,,33 34 ,,35 36 , 3
22 32
,, 42 52
,, 62 

21 31
,, 41 51
,, 61 
26
36
,,
46
2
1

Note that and means the intersection of the sets.
36 18
-5-
56
,,
Unit 1: Probability
2. The event that one of the dice shows a three or the sum is 8;
,,
16
15
,,

14
,,

,,
13
12
,,

,,
11
P (3  8) 
66 
25 35
,, 45 55
,, 65 

24 34
,, 44 54
,, 64 

2,,33 34 ,,35 36 , 3
22 32
,, 42 52
,, 62 

21 31
,, 41 51
,, 61 
26
36
,,
46
56
,,
13
Note that or means the union of the sets.
36
In this case the two events (showing a three and sum of 8) are not mutually exclusive
because the elements (5, 3) and (3, 5) are part of both of the events.
If two events A and B are not mutually exclusive, then
P(A  B) = P(A) + P(B) - P(A  B)
In the previous example the probability of one 3 or a sum of 8 is;
P(3  8) = P(3) + P(8) - P(3  8)
10 5
2 13

 

36 36 36 36
In expanded form we could say; The event that the dice show a 3 or have a sum of 8 is equal
to the event of a 3 plus the event of a sum of 8 subtract the event that it shows a 3 and a sum
of 8.
Mutually exclusive events have no part in common.
For Example; The event where one of the dice shows a 3 and the sum is 10.
,,
16
15
,,

14
,,

,,
13
12
,,

,,
11
66 
25 35
,, 45 55
,, 65 

24 34
,, 44 54
,, 64 

2,,33 34 ,,35 36 , 3
22 32
,, 42 52
,, 62 

21 31
,, 41 51
,, 61 
26
36
,,
46
56
,,
P(3  10)  
The event where one of the dice shows a 3 or the sum is 10.
,,
16
15
,,

14
,,

,,
13
12
,,

11
 ,,
66 
25 35
,, 45 55
,, 65 

24 34
,, 44 54
,, 64 

2,,33 34 ,,35 36 , 3
22 32
,, 42 52
,, 62 

21 31
,, 41 51
,, 61 
26
36
,,
P (3  10) 
46
10 3 13


36 36 36
-6-
56
,,
Unit 1: Probability
Events A and B are mutually exclusive iff (if and only if);
P( A  B)  P( A)  P( B)
P( A  B )  
In the previous example the probability of one 3 or a sum of 10 is;
P (3  10)  P (3)  P (10)
10 3 13



36 36 36
In expanded form we could say; The events that the dice show a 3 or have a sum of 10
are mutually exclusive because their union is equal to the sum of their individual events.
When two or more events are considered in sequence, it is necessary to determine if they are
Dependent or Independent Events.
In Dependent Events, the first event affects the probability of the second.
For example: when drawing two cards from a standard deck, if the cards are not replaced
between drawings, the absence of the first card will affect the probability when drawing a
second.
Independent Events do not affect each other. In the example above the events could be
made independent if the card is replaced in the deck after drawing.
We can calculate the probability of successive events A and B occurring;
P(A  B) = P(A)•P(B|A)
Where P(B|A) is the probability of event B occurring after event A
If A and B are Independent then P(B|A) = P(B) and;
P(A  B) = P(A)•P(B)
Examples:
When drawing two cards in succession from a standard deck what is the probability that two
deuces will be drawn if;
a) The cards are replaced and the deck is shuffled between drawings?
Solution: Since the cards are replaced
P ( deuce  deuce)  P ( deuce)  P ( deuce)
4 4
1 1
1
   
52 52 13 13 169
b) The cards are not replaced after drawing?

Solution: Since the cards are not replaced
P ( deuce  deuce)  P ( deuce)  P ( deuce | deuce)

4 3
1 1
1
   
52 51 13 17 221
-7-
-
Unit 1: Probability
Practice Questions 2: Probability
I. Mutually Exclusive Events
1. Two dice are rolled. What is the probability that:
a. the sum of the numbers is 4.
b. one or both of the numbers is 6.
c. either one of the numbers is 4 or one or both of the numbers is 6.
2. A single card is drawn from a standard deck, what is the probability that:
a. it is a club or a diamond?
b. it is a heart or a black jack?
c. it is a red ace or a black face card.
d. it is a heart or a 4?
e. it is a diamond or a face card?
f. It is neither an ace nor a spade?
3. Two dice are rolled. What is the probability that either the sum of the numbers is 5 or one of
the numbers is a 6?
4. Two dice are rolled. What is the probability that either the sum of the numbers is 5 or one of
the numbers is a 2?
5. Why are the answers to 4. and 5. different?
6. A box contains 3 blue balls, 6 green balls, 4 red balls, and 5 black balls. One is drawn at
random. Find the probability of choosing:
a. a blue ball.
b. a green ball.
c. a green or a red ball
d. a blue or a green or a black ball.
e. not a red ball.
f. why are d. and e. the same?
7. A coin is tossed three times. Find the probability of:
a. 3 heads or three tails.
b. 2 or three heads.
c. 2 or 3 heads or all tosses the same.
II. Dependent and independent events
8. Find the probability of:
a. tossing a head with 1 coin.
b. tossing a tail with 1 coin.
c. tossing a head and a tail with 2 coins.
9. Cards are drawn from a well shuffled standard deck of cards. They are picked one at a time,
and replaced before picking the next one. Find the probability of:
a. picking a king.
b. picking 2 kings.
c. picking 2 clubs.
d. picking a heart and a club.
10. Three cards are dealt from a well shuffled standard deck. Find the probability that:
a. all the cards are tens
b. all the cards are spades.
11. What is the probability of tossing 8 heads in a row with a coin?
12. If a coin comes up heads 8 times in a row, what is the probability of a head on the ninth
toss?
-8-
Unit 1: Probability
13. A box contains 3 black balls, 4 blue balls, and 5 green balls. A ball is selected at random,
replaced and another is selected. Find the probability that;
a. the first is green and the second is blue. b. both balls are black
c. both balls are blue
d. the first is black and the second is green.
e. the first is green and the second is black.
14. If three coins are tossed find the probability of tossing three heads.
15. If the probability that Stacey will pass Math B30 is 5/6, that David will pass is 1/4, and that
Kelley will pass is 3/5. Find the probability that:
a. all three pass
b. only Stacey passes
c. only David Fails
d. David and Kelley pass.
III: Anything goes.
16. Two dice are rolled. What is the probability that:
a. one number is a 6?
b. the sum of the numbers is 5?
c. one of the numbers is 6 or the sum is 5?
d. at least one number is a 2?
e. the sum of the numbers is 7?
f. at least one of the numbers is 2 or the sum is 7?
g. at least one of the numbers is 2 and sum is 7?
17. If two cards are drawn from a standard deck without replacement, what is the probability
that:
a. both are black?
b. one is the ace of spades?
c. either both are black or one is the ace of spades?
18. A coin is tossed three times.
a. What is the probability that the first two tosses produce heads?
b. What is the probability that the last toss produces a tail?
c. Are the events in a. and b. independent?
d. What is the probability that at least two tosses produce heads?
e. What is the probability that at least one toss produces a tail?
f. Are the events in d. and e. independent?
19. One bag contains 3 red, 2 black, and 2 white balls. Another bag contains 1 red, 1 black, and 3
white balls. If a ball is drawn at random from each bag, find the probability that either both
are black or neither is black.
20. If the probability that John will solve a certain problem is 2/3, that Mary will solve it is 3/4,
and that Bill will solve it is 1/2, what is the probability that:
a. at least one person solves it?
b. Mary and Bill will solve it but John will not?
c. John and Mary will solve it but Bill will not?
d. all of them solve it?
e. None of them solve it?
-9-
Unit 1: Probability
Binomial Theorem and Binomial Probability
Objectives:
A.6
To determine the coefficients of the terms in a binomial expansion using the Binomial
theorem.
A.7
To expand binomials of the form (a + b)n using the Binomial theorem.
A.8
To solve word problems associated with Objectives A.6 and A.7.
To use Binomial theorem to find probabilities for succesive trials of events with 2 possible
outcomes.
Notes:Introduction to Binomial Theorem
Calculate the following;
(x + y)2 = (x + y)(x + y) = x2 + 2xy + y2
Remember to use F.O.I.L.
(x + y)3 = (x + y)(x2 + 2xy + y2)
We can use the answer to the previous one.
3
2
2
2
2
3
= x + 2x y + xy + x y + 2xy + y Use the distributive property.
= x3 + 3x2y + 3xy2 + y3
Simplify
(x + y)4 = (x + y)(x3 + 3x2y + 3xy2 + y3)
= … = x4 + 4x3y + 6x2y2 + 4xy3 + y4
Again, using the previous answer.
Multiplied and simplified.
From the examples above, it is possible to notice several trends in the expansion of powers of
binomials;
- The exponents in each term add to equal the exponent on the original question.
- The coefficients in each expansion seem to follow some pattern;
Binomial
(x + y)0 =
(x + y)1 =
(x + y)2 =
(x + y)3 =
(x + y)4 =
(x + y)5 =
Expanded
1
1x + 1y
1x2 + 2xy + 1y2
1x3 + 3x2y + 3xy2 +1 y3
1x4 + 4x3y + 6x2y2 + 4xy3 + 1y4
5
1x + 5x4y + 10x3y2 +10x2y3 + 5xy4 + 1y5
Coefficients
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
The triangular array on the right is known as Pascal’s Triangle [Blaise Pascal(1623 - 1662)].
The members of each row are generated by adding the two numbers in the row above.
In the previous example, the numbers in the bottom row were found by adding;
1
1
4
5
6
10
4
10
1
5
1
In general we can find the coefficients for a binomial expansion of the form (a + b)n by
finding the numbers in the (n + 1)th row of Pascal’s triangle. This is made even easier if we
notice that the second number in the row is = n.
-10-
Unit 1: Probability
For Example:
(x -2y)4 = 1x4 + 4x3(-2y) + 6x2(-2y)2 + 4x(-2y)3 + 1(-2y)4
Coefficients from
= x4 + 4x3(-2)y + 6x2(-2)2y2 + 4x(-2)3y3 + 1(-2)4y4 the fifth row of
= x4 - 8x3y + 24x2y2 - 32xy3 + 16y4
Pascal’s Triangle
Practice Questions 3: Pascal’s Triangle
1. Copy Pascal’s Triangle into your book and fill in the blank spaces up to the tenth row.
1
1
1
1
1
2
3
1
3
1
1
4 6 4 1
5 10 10 5 1
1 6 ___ ___ ___ 6 1
1 ___ ___ ___ ___ ___ ___ 1
etc.
1
2. Use Pascal’s Triangle to expand each of the following completely;
a. (x + y)7
c. (2x + 1)5
x

e.   3 
3

b. (x - y)5
d. (1 + 3y)4
6
Y

f.  2  
2

Use the variables x and (-y)
8
Binomial Expansion Using Combinations
Using Pascal’s Triangle is less useful if we wish to expand larger powers of binomials.
For example (x + y)25 would require a triangle with 26 rows. The rows of Pascal’s Triangle
can be generated using the formula for combinations ;
n
Cr 
n!
, where n is the exponent of the binomial and r + 1 is the number of the term.
(n  r )!r !
The general formula for the expansion of a binomial of the form (a + b)n is;
 a  b
n
 n C0 a nb0  n C1a n 1b1  n C2 a n  2b 2 
 n Cn a 0 b n
For Example:
 x  y
25
 25 C0 x 25 y 0  25 C1 x 24 y1  25 C2 x 23 y 2 
25! 25
25! 24 1 25! 23 2
x 1 
x y 
x y 
25!
24!
23!2!
 1x 25  25 x 24 y  300 x 23 y 2   1y 25

-11-
 25 C25 x 0 y 25

25! 0 25
x y
25!
Unit 1: Probability
TI-82 Calculator: to evaluate an expression using n Cr enter the following;
n ,[MATH], [→] (arrow to PRB menu), 3 (select nCr), r
Example 25C2 →Enter: 25 [MATH] [→] [→] [→] 3 2 [ENTER] Result: 300
Example: List the first 4 terms of the expansion of (2 - x)18
Solution:
2  x
18
 18 C0 (2)18 ( x) 0  18 C1 (2)17 ( x)1  18 C2 (2)16 (  x) 2  18 C3 (2)15 (  x) 3 
 1 262144 1  18 131072  (  x)  153  65536  (  x) 2  816  32768  (  x) 3 
 262144  2359296 x  10027008 x 2  26738688 x3 
Notice that the coefficients can become very large. Also note the effect of a negative
coefficient within the binomial.
n
**In some texts n Cr may be expressed as;  
r 
Practice Questions 4: Binomial Expansions using combinations.
1. List the first four terms in each of the following;
a. (a + b)20
b. (a - b)12
18
c. (x - 2y)
d. (t2 + 1)11
2. Write the value of n for which each term is a term of the expansion of (p + q)n
a. p7q2
b. p4q4
2 5
c. p q
d. p7q
e. p8q8
3. Write the value of n in each term of the expansion (p + q)12
a. p3qn
b. pnq7
c. pqn
d. pnq
12 n
e. p q
4. Calculate 1013 by writing it as (100 + 1)3 and expanding.
Using the Formula for the rth Term
Another method for finding particular terms in an expansion is to use a general formula for
the “rth” term of the binomial expansion;(a + b)n:
tr  n Cr 1a ( n  r ) 1b r 1 r
-12-
Unit 1: Probability
For Example: Find the 6 term of the expansion (x + 2y)14.
Solution:
t6  14 C61 x (146) 1 (2 y )61
th
 14 C5 x 9 (2 y )5
14! 9 5 5
x 2 y
9!5!
87178291200 9

x  32  y 5
362880 120
87178291200

 32 x9 y 5
43545600
 2002  32 x9 y 5  64064x 9 y 5

Example: Find and simplify the term with s5 in the expansion (3s - t)8
Solution: The exponent for the first variable in each term is (n-r)+ 1
 5 = (8 - r) + 1  r = 4
t4  8 C41 (3s )(8 4) 1 (t ) 41
8!
(3s)5 (t )3
3!5!
40320 5 5

3 s (1)3 t 3
6 120
 56  243s5 (1)t 3  13608s5t 3

Practice Questions 5: Terms of a Binomial Expansion
1. Find and simplify the specified term in each expansion;
a. The fifth term of (x + y)10
b. The sixth term of (a + b)11
8
b

c. The fourth term of  2  
2

e. The middle term of (x2 + y2)6
g. The term with x3 in (a + x)8
10
b

d. The eighth term of  2  
2

f. The middle term of (c2 - 2d)8
h. The term with y4 in (a - y)9
2. In the expansion of (a + b)38, one term of the expansion is akb15:
a. What is the value of k?
b. What is the numeral coefficient?
3. What is the coefficient of xy4 in each of the following?
a. (x - y)5
b. (x + 2y)5
5
c. (2y - x)
d. (3x + y)5
-13-
Unit 1: Probability
Binomial Probability
In repeated trials of an event with only two possible outcomes, we can use a binomial
expansion to find the probability of achieving any combination of outcomes.
For example consider the tossing of a coin; the two possible outcomes are heads(H) and
tails(T). Since they are equally probable, P( H )  P(T )  12 . All possible combinations for the
results of 5 tosses of a coin can be found using the expansion (h + t)5.
(h  t )5  5 C0 h5t 0  5 C1h4t1  5 C2 h3t 2  5 C3h2t 3  5 C4h1t 4  5 C5h0t 5
 1h5  5h4t  10h3t 2  10h2t 3  5ht 4  t 5
If we let h  P( H )  12 and t  P (T )  12 , then each term can be used to calculate the
probability of the corresponding outcome;
The probability of tossing 4 heads and 1 tail can be found with the term;
4
5h 4t  5  12   12   5  161   12   325
Example: If a coin is tossed 20 times, find the probability of tossing 5 heads and 15 tails.
Solution: This situation is described by (h + t)20. The term we want is the one with h5t15,
which will be the 16th term. (The exponent on the second variable = r-1)
t16  20 C161h (2016) 1t161
 15504h5t15
 15504 
 
1 5
2
1 15
2
15504
 15504
 1048576

220
Let h  P( H )  12 and t  P (T ) 
1
2
969
65536
Example: A single die is rolled 10 times, what is the probability of getting three 3’s?
Solution: In this situstion we can let t  P(3)  16 and n  P(3)  56 . (Remember that 3 means
not 3)
This event can be represented as (t + n)10. The term we want is the one with t3n7 or the 8th
term.
t4  10 C81t (108)1n81
 10 C7t 3 n7
 120  16 

3
 56 
Let t  P(3)  16 and n  P(3)  56
7
120  78125 9375000
390625


60466176 60466176 2519424
-14-
Unit 1: Probability
Practice Questions 6: Binomial Probability
1. Find the probability of tossing 2 heads and a tail with 3 tosses of a coin.
2. Find the probability of tossing 5 heads and 2 tails with 7 tosses of a single coin.
3. Calculate the possibility of the following:
a. 4 heads in 7 tosses of a coin.
b. 5 heads in 7 tosses of a coin.
c. 6 heads in 7 tosses of a coin.
d. 7 heads in 7 tosses of a coin.
e. at least 4 heads in 7 tosses of a coin.
Note: “at least” means 1, or 2, or 3, etc.
4. A die is tossed 6 times and you are to find the number of times that a two occurs.
a. The binomial expansion (x + y)6 can be used to solve the problem involving six tosses
of a die. What value will you give x, and y?
b. Which term and coefficient that will give the probability of tossing a two three times?
c. calculate the probability of tossing a two three times in six tosses of a single die.
5. Calculate the probability of obtaining 4 heads in ten tosses of a single coin.
6. Find the probability of rolling five 3’s in 15 tosses of a single die.
7. A multiple choice test has 8 questions. Each question has four choices.
a. what is the probability of guessing the correct answer to a single question?
b. What is the probability of guessing the correct answer to exactly two questions?
c. What is the probability of guessing the correct answer to exactly three questions?
d. What is the probability of guessing the correct answer to at least three questions?
e. What is the probability of guessing the correct answer to the first two questions and
incorrect on the rest of the test?
-15-
Unit 1: Probability
Unit 1: Solutions
Practice Questions 1: Union and Intersection of Sets Page 2
1. a. T b. F c. T d. T
2. a. {3, 5} b. {2, 3, 4, 5, 7, 9} c. {2, 3, 4, 5, 7, 9} d. {2, 3, 4, 5, 7, 9}
3. a. {1} b. {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15} c. {1, 2, 3, 4, 5, 11}
4. a. 12
b. 20
c. 6
d. 10
e. 38
5.a.
Q
SIFC
English
S
Science
8
12
10
6
8
T
R
10
b.
6
Q
S
History
20
T
R
6. a. 62
b. 12
c. 35
c.
d. 249
Q
Grade 12
Math
S
Physics
34
62
40
27
T
R
12
22
d.
Q
35
S
Chemistry
17
R
T
7. 68
Practice Questions 2: Probability Page 8
1. a. 121 b. 11
c. 187
2. a. 12 b. 175 c. 132 d. 134 e. 11
f. 139
36
26
3. 125 4. 13
5. Question 3 describes mutually exclusive events, question 4 does not.
36
6. a. 16 b. 13 c. 95 d. 79 e. 79 f. The events discribed in d, and e are the same.
-16-
Unit 1: Probability
1
7. a.
b.
c.
8. a.
b.
c. 12 9. a. 131 b. 169
c. 161 d. 161
1
11
1
10. a. 5525
b. 850
11. 256
12. 12 13. a. 365 b. 161 c. 19 d. 485 e. 485
1
4
1
2
5
8
1
2
1
2
14. 18 15. a. 18 b. 14 c. 83 d. 203 16. a. 185 b. 19 c. 187 d. 11
e. 16 f. 15
g.
36
36
25
9
7
1
1
1
1
17. a. 102 b. 26 c. 34 18. a. 4 b. 2 c. yes d. 2 e. 8 f. no
23
22
19. 35
20. a. 24
b. 18 c. 14 d. 14 e. 241
1
18
Practice Questions 3: Pascal’s Triangle Page 11
1.
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1
1 8 28 56 70 56 28 8 1
1 9 36 84 126 126 84 36 9 1
1 10 45 120 210 252 210 120 45 10 1
2. a. (x + y)7 = x7 + 7x6y + 21x5y2 + 35x4y3 + 35x3y4 + 21x2y4 + 7xy6 + y7
b. (x - y)5 = x5 - 5x4y + 10x3y2 - 10x2y3 + 5xy4 - y5
c. (2x + 1)5 = 32x5 + 80x4 + 80x3 + 40x2 + 10x + 1
d. (1 + 3y)4 = 1 + 12y + 54y2 + 108y3 + 81y4
6
4
x6
x5
 81
 x9  x3  9 x 2  81x  729
e.  3x  3  729

f. 2  2y

8
 256  64 y  16 y 2  4 y 3  y 4 
y5
4
6
7
8
y
y
y
 16
 64
 256
Practice Questions 4: Binomial Expansions Using Combinations Page 12
1. a. (a  b)20  a 20  20a19b  190a18b2  1140a17b3 
b. (a  b)12  a12  12a11b  66a10b2  220a9b3 
c. ( x  2 y)18  x18  36 x17 y  612 x16 y 2  6528x15 y 3 
d. (t 2  1)11  t 22  11t 20  55t18  165t16 
2. a. 9 b. 8 c. 7 d. 8 e. 16 3. a. 9 b. 5 c. 11 d. 11 e. 0
4. (100  1)3  1003  3 1002 1  3 100 12  13  1,000,000  30,000  300  1  1,030,301
Practice Questions 5: Terms of a Binomial Expansion Page 13
1. a. t5  4845 x16 y 4 b. t6  462a 6b5 c. t4  1920b3 d. t8  152 b7 e. f. t5  1120c8 d 4 g.
t5  56a 5 x3 h. t5  126a 5 y 4
2. a. k = 23 b. 1.547128656 × 1010
3. a. 5
b. 80
c. -80
Practice Questions 6: Binomial Probability Page 14
35
7
21
21
1
1. e 83 2. 128
3. a. 128
b. 128
c. 128
d. 128
e. 12
625
4. a. x  16 , y  56 b. t4  20 x 3 y 3 c. 11664
1701
5103
729
7. a. 14 b. 16384
c. 8192
d. 21067
65536 e. 65536
5. 105
512
6. 0.06237
-17-
d. 15
Unit 1: Probability
Unit 1: Review
1. A SET is any well defined collection of items. Each element within the set is called a
MEMBER of that set.
A set may be designated by name, by listing the members, by definition, or with a Venn
Diagram. Example; A = {0, 2, 4, 6…},A = {x | x  W, x is a multiple of 2}
The set “A” is given by list and by the definition that; A is the set of all numbers, x, such
that x is a member of the set of whole numbers and x is a multiple of 2.
Definitions and lists are enclosed in curly brackets.{}
The symbol “” means “is a member of”.
The symbol “|” within a definition means “such that”.
The empty set is Ø or {} but never {Ø}.
2. The UNION() of two sets includes all of the members in both sets.
The INTERSECTION() of two sets includes all of the members common to both sets.
AB
A
AB
B
A
B
AB=A+B-AB
3.
The PROBABILITY of an event can be found using the relation;
favorable outcomes
event space
P(a) 

total possible outcomes sample space
The SAMPLE SPACE is the set of all possible outcomes.
An EVENT is one or more possible outcomes and the EVENT SPACE is its set.
Sample spaces and Events can be calculated using principles of permutations and
combinations developed in Mathematics A30.
4. The probability that A or B will occur is given by;
P(A)  P(B) = P(A) + P(B) - P(A  B)
If P(A  B) = Ø (the sets have nothing in common), then A and B are MUTUALLY
EXCLUSIVE and P(A)  P(B) = P(A) + P(B)
5. The probability that A and B will occur is given by;
P(A)  P(B) = P(A)•P(B|A)
Where P(B|A) is the probability of event B after event A has occurred.
If event A changes the probability of event B, then A and B are DEPENDENT events
and P(B) ≠ P(B|A).
-18-
Unit 1: Probability
If event A has no effect on the probability of event B, then A and B are INDEPENDENT
events and P(B) ≠ P(B|A).  P(A)  P(B) = P(A)•P(B)
6. The coefficients of a BINOMIAL EXPANSION,  a  b  , can be found using Pascal’s
Triangle or by;
n
 a  b  n C0 a nb0  n C1a n1b1  n C2 a n2b 2   n Cn a 0b n
n
7. The rth term of a binomial expansion ,  a  b  , can be found using the formula;
n
tr  n Cr 1a ( n  r ) 1b r 1
8. The terms of a binomial expansion can be used to find the probabilities of repeated trials of
events with two possible outcomes.
Unit 1 Review Questions
1. Five married couples sit at a circular table. In how many ways can they be seated if a woman
must sit next to a man?
2. Given A = {3, 4, 5, 6, 7}, B = {4, 5, 7, 14, 22}, and C = {1, 2, 8, 9}. Find the following:
a. A  B
b. A  C
c. A  B  C
d. A  C
3. Draw a Venn Diagram to represent the following:
U = {5, 6, 7, 8, 9, 10, 11, 12, 13}, M = {5, 8, 10, 11}, and N = {5, 6, 7, 9, 10}
4. The probability that Bob will ask Julie to a movie is 14 , that Jack will ask her is 12 , and that
Tom will ask her is 18 . Find the probability that:
a. Only Bob and Jack ask her.
b. Only Jack asks her.
c. All three ask her.
d. None of them ask her.
5. Two cards are drawn from a standard deck. Find the probability that:
a. the first is a queen and the second is black if the first card is replaced before the second
is drawn.
b. the cards are both queens if the first card is not replaced.
c. they are both fives or both red. No replacement.
d. they are both queens or both black. No replacement.
e. they are both jacks or both kings. No replacement.
6. Which of the following pairs of events are independent?
a. Throwing a three and a five on succesive rolls of a die.
b. Studying for a math B30 exam and passing it.
c. Not having a date and going to a dance.
d. Throwing heads on a toss of a coin and throwing heads on the first two of three tosses.
e. Being blond and wearing glasses.
f. Having a good coach and winning the regional title.
g. Being the top student in math and the top student in science.
-19-
Unit 1: Probability
h. Throwing tails on 50 tosses of a coin and throwing tails on the 51st toss.
i. Winning at high jump and wearing steel toed shoes.
j. Playing high school volleyball and playing badminton.
7. Which of the following pairs of events are mutually exclusive:
a. Drawing a king, drawing a queen.
b. Drawing an ace, drawing a heart.
c. Drawing a face card, drawing the king of spades.
d. Drawing a face card, drawing the 6 of hearts.
e. Having blue eyes, having brown hair.
f. Having blue eyes, wearing glasses.
g. It is raining, it is monday.
h. Joe lives in Canada, Joe lives in Moose Jaw.
i. Being born in Moose Jaw, being born in Ottawa.
j. It is Friday, having math class.
8. Two dice are rolled. What is the probability that the sum of the numbers is four or at least
one of the numbers is a one?
9. Expand the following completely:
a. (2x - 3)4
b.
10. Find the following terms:
a. the 4th term of (2a - 10)10
b. the middle term of
 4x  4 
5
 2x  2 
6
11. Using binomial probability, find the probability of tossing 6 heads with 6 tosses of a single
coin.
12. A single card is drawn from a standard deck. What is the probability that the card is:
a. a spade or a heart?
b. a spade or a red jack?
c. a black ace or a red face card?
d. a club or a four?
e. a heart or a face card?
f. neither a king nor a heart?
13. Three cards are drawn from a standard deck in succession and each is replaced after drawing.
Find the probability that:
a. all three cards are aces.
b. all three cards are diamonds.
c. all three cards are the same suit.
d. a diamond, a heart, and a club are drawn in order.
e. a five, a diamond and a red card are drawn in order.
f. a five, a diamond and a red card are drawn in any order.
14. A jeweller has 15 different sized pearls to string on a necklace.
a. How many ways can this be done?
b. If three pearls are the same size, how many ways can this be done?
-20-
Unit 1: Probability
Unit 1 Review Solutions
1. 5760
2. a. {3, 4, 5, 6, 7, 14, 22}
d. {1, 2, 3, 4, 5, 6, 7, 8, 9}
3. Diagram:
U
11
8
12
5
10
M
b. Ø
c. {1, 2, 3, 4, 5, 6, 7,8, 9, 14, 22}
4. a.
6
N
7
9
17
64
5. a.
13
b.
21
64
c.
1
64
d.
21
64
1
1
140
140
2
b.
c.
d.
e.
26
221
663
663
221
6. a. independent b. dependent c. dependent d. dependent e. independent f. dependent
g. dependent h. independent i. dependent j. independent
7. a. exclusive b. not exclusive c. not exclusive d. exclusive e. not exclusive f. not exclusive
g. not exclusive h. not exclusive i. exclusive j. not exclusive
8.
1
3
9. a. 16 x 4  24 x3  36 x 2  54 x  81 b.
10. a. -15360000a7 b. -20x3
13. a.
1
2179
b.
1
64
c.
1
16
d.
1
64
11. 641
e.
1
104
12. a.
3
f. 104
x5
1024
 564x  52x  40 x2  320 x  1024
1
2
b.
3
15
52
c.
2
13
d.
4
13
e. 11
26
f.
9
13
14. a. 43,589,145,600 b.7,264,857,600
-21-
4
Unit 1: Probability
-22-