Download Probabilities

20
◙ EP-Program
- Strisuksa School - Roi-et
Math : Probabilities
► Dr.Wattana Toutip - Department of Mathematics – Khon Kaen University
© 2010 :Wattana Toutip
◙ [email protected]
◙ http://home.kku.ac.th/wattou
20. Probability
20.1 Probability of successive events
A Tree Diagram shows the probability of successive events
20.1.1 Examples
1. A bag contains 5 red and 6 black marbles. Two
are drawn without replacement. What is the
probability that :
(a) Both are red
(b)
They are different colors.
Solution
Construct the diagram as shown. The top branch
corresponds to both marbles being red. Multiply the
probabilities along this branch.
Fig 20.1
5 4 2
(a) The Probability that both are red is  
11 10 11
The two middle branches correspond to the
marbles having different colors.
5 6 6 5 6
(b) P(different colors)     
11 10 11 10 11
2. Four fair dice are rolled. Find the probabilities of
(a) Four 6s
(b) at least one 6.
Solution
4
1
(a) The Probability of four 6s is  
6
1
The Probability of four 6s is
1296
4
5
(b) There will be at least one 6 unless none of the dice show a 6. This has probability   .
6
625
671

The probability of at least one 6 is 1 
1296 1296
20.1.2 Exercises
1. Two fair dice are rolled. Find the probabilities that:
(a) Both are sixes
(b) The total is 2
(c) The total is 7
(d) The first is greater than the second
(e) A 'double' is thrown
(f) At least one of the dice is a six.
2. Two card are drawn without replacement from a well-shuffled pack. Find the probabilities
that :
(a) The first is a heart
(b) Both are heart
(c) The first is a heart and the second is a spade
(d) The first is a King and the second is a Queen.
3. A bag contains 5 blue are 4 green counters. Two are drawn without replacement. Find the
probabilities that :
(a) Both are blue
(b) They are the same colour
(c) There is at least one blue
(d) The second is a green.
4. A sweet box contains 5 toffees, 6 liquorices and 8 chocolates. Two are drawn out. Find the
probabilities that :
(a) The first is a toffee and the second is a chocolate
(b) At least one is a liquorice
(c) Neither is a toffee.
5. Two fair dice are rolled. The score is the larger of the numbers showing. Find the
probabilities that:
(a) The score is 1
(b) The score is 6
6. Two children A and B each pick at random a single digit from 1 to 9. Find the probabilities
that :
(a) They pick the same number
(b) A's number is larger than B's
7. To start a certain board game a die is rolled until a six is obtained. Find the probabilities
that :
(a) A player starts on his first roll
(b) He starts on his second roll
(c) He starts on his third roll
(d) He has not started by his fourth roll
1
8. To start at darts a' double' must first be thrown. Albert has probability
of throwing a
10
1
double, and Beatrice has probability . Albert throws first. Find the probabilities that :
8
(a)Both start on their first throw
(b) Beatrice starts on her second throw but Albert has not started by then.
9. A fair coin is spun five times. Find the probabilities of
(a) five Heads
(b) at least one Head.
10. A roulette wheel has the number 1 to 36. A gambler bets that a number divisible by 3 will
turn up. The bet is repeated four times. Find the probabilities that the gambler
(a) Wins all his bets (b) wins at least one bets.
11. Five people take the driving test. Each has probability of passing. Find the probabilities
that :
(a) They all pass
(b) at least one fails.
20.2 Exclusive and independent events. Conditional probability
If two events cannot happen together, then they are exclusive. If events are exclusive
then the probability that one or the other occurs is the sum of their probabilities.
P  A or B   P  A   P  B  , provided that A and B are exclusive.
Two events are independent if the truth of the one of them does not alter the probability of the
other.
If events A and B are independent then the probability of them both occurring is the product
of their probabilities.
P  A & B  P  A  P B
The conditional probability of 6!  720 A given B is the probability of A , once it is known
that B is true.
The conditional probability of A given B is written P  A | B  . It is obtained from the formula
:
P  A | B 
P  A & B
P B
If A and B are independent then P  A | B  P  A 
The symbols  and  are sometimes used for 'and' and 'or' respectively.
20.2.1 Example
1. Two cards are drawn without replacement from a pack. Events A, B, C are as follows :
C : the card is a king.
A : the first is a heart.
B : the second is a heart.
Which pairs of these events are independent?
Solution
12 1
If A is true, then there is one fewer heart in the pack. The probability of B is  . Hence
51 4
A and B are not independent.
1
If A is true then C is neither more likely nor less likely than before. Its probability is still .
13
Hence A and C are independent.
Similarly the truth of B does not alter probability of C .
The pairs A and C, B and C are independent
2. A women travels to work by bus, car or on foot with probabilities
1 1 1
, , respectively.
6 3 2
1 1 1
, ,
respectively. If she
5 4 20
arrives late one morning, find the probability that she come by bus.
Solution
For each type of transport her probabilities of being late are
Here let L be the event that she is late, and B the event that she came by bus. Use the formula
for conditional probability:
P ( B & L)
P  B | L =
P ( L)
1 1

6
5
P ( B | L) 
1 1 1 1 1 1
    
6 5 3 4 2 20
4
The probability that she came by bus is
17
20.2.2 Exercises
1. Two cards are drawn without replacement from a pack. Events A, B,C,D are as follows :
A : the first is an ace.
B : the second is an 8.
C : the first is red.
D: the second is an spade
Which pairs of events are independent? Which are exclusive?
2.Two dice are rolled. Events A, B, C, D are as follows:
A : the first is an 5
B : the total is 8
C : the total is 7
D: the dice show the same number
Which pairs of events are independent? Which are exclusive?
3. Two dice are rolled. Events A, B, C, D are as follows :
A : the total is 7
B : the second die is a 2
C : both dice are less than 5 D: at least one die is a 6
Which pairs of events are independent? Which are exclusive?
4. Three fair coins are spun. Events A, B, C, D are as follows :
A : the first coin is a head
B : all the coins are heads
C : there is at least one tail D: the first and last coins show the same
Which pairs of events are independent? Which are exclusive?
5. With A, B, C, D as defined is Question 1, Find the following:
(a) P  B | A 
(b) P  D | C 
6. With A, B, C, D as defined is Question 2, Find the following:
(a) P  A | B 
(b) P  B | D 
7. With A, B, C, D as defined is Question 3, Find the following:
(a) P  A | B 
(b) P  C | D 
8. With A, B, C, D as defined is Question 4, Find the following:
(a) P  B | A 
(b) P  C | A 
9. A box contains 5 red and 6 blue marbles. Two are drawn without replacement. If the
second is red find the probability that first was blue.
10. In his drawer a man has 7 left shoes and 10 right shoes. He picks two out at random.
Find the probability that:
(a) He has one left shoe and one right shoe
(b) He has one left shoe and one right, given that the first was a left
(c) The second is a left, given that he has one has one left and one right.
11. A man travels to work by bus, car and motorcycle with probabilities 0.4, 0.5, 0.1
respectively. With each type of transport his chances of an accident are
1 1 1
, ,
500 50 10
respectively. Find the probabilities that:
(a) He goes by car and has an accident.
(b) He does not have an accident.
(c) He went by motorcycle, given that he had an accident.
12. 1% of the population has a certain disease. There is a test for the disease, which gives a
9
1
positive response for
of the people with the disease, and for
of the people without the
10
50
disease. A person is selected at random and tested.
(a) What is the probability that the test gives a negative response?
(b) If the test is positive, what is the probability that the person has the disease?
(c) If the test is negative, what is the probability that the person does not have the disease?
2
13. An island contains two tribes;
of the population are Wache , who tell the truth with
3
1
probability 0.7 , and
are Oya, who tell the truth with probability 0.8 . I meet a tribesman
3
who tell me that he is a Wache. What is the probability that he is telling the truth?
3
14. In the certain town
of the voters are over 25 , and they vote the Freedom Party with
4
1
probability . Voters under 25 support the Freedom Party with probability . If a support of
3
the Freedom Party is picked at random, what is the probability that he or she is under 25?
15. Events A and B are such that P  A   0.4, P  B   0.3 , and P  A & B   0.25 .
Show that A and B are neither exclusive nor independent. Find P  A | B  .
16. Events A and B are such that P  A   0.3, P  B   0.2 , and P  A or B   0.4 . Show that A
and B are not exclusive. Find P  A & B  and P  A | B  .
17. Events A and B are such that P  A   0.4, P  B   0.3 , and P  A | B   0.5 . Find P  A & B 
and P  A or B Find P  B | A 
18. Events A and B are such that P  A   0.3, P  B   0.5 . Find P  A or B and P  A & B  in
the following cases:
(a) A and B are exclusive
(b) A and B are independent
20.3 Examination questions
1. A 2p coin a 10p are throw on a table. Event A is 'A head occurs on the 2p coin'. Event B is
'A head occurs on the 10p coin'. Event C is 'Two head or two tails obtained'.
State, giving reasons, which of the following statements is (are) true and which is (are) false.
(a) A and B are independent events
(b) B and C are independent events.
(c) A and C are mutually exclusive events
(d) A and BC are independent events.
(e) P  A  B  C  = P  A  .P  B  .P  C 
2. Six balls colored yellow, green , brown, blue, pink , and black have values
2, 3, 4, 5, 6, 7 respectively. They are independent in size and placed in a box. Two balls are
selected together from the box at random and the total number of point recorded
(i) Find the probability that the total score is
(a) 7 , (b) 9 , (c) 10 , (d) greater 9 , (e) odd.
(ii) A game between two players, X and Y , starts with the six balls in a box. Each
player in turn selects at random two balls, notes the score and then returns the balls to
the box. The game is over when one of the players reaches a total score of 25 or more.
(a) If X starts, calculate the probability that X wins on his second turn;
(b) If Y starts, calculate the probability that Y wins on his second turn.
[O ADD]
3. (a) The two electronic systems C1 , C2 of a communications satellite operate independently
and have probabilities of 0.1 and 0.05 respectively of failing. Find the probability that
(i) neither circuit fails
(ii) at least one circuit fails,
(iii) exactly one circuit fails
(b) In a certain boxing competition all fights are either won or lost; draws are not
permitted.
3
If a boxer wins a fight then the probability that he wins his next fight is ; if he loses
4
2
a fight the probability of him losing the next three fights is .
3
Assuming that he won his last fight, use a tree diagram, or otherwise, to calculate the
probability that of his three fights
(i) he wins exactly two fights
(ii) he wins at most two fights.
State the most likely and least likely sequence of results for these three fights.
4. (i) The events A and B are such that P  A   0.4, P  B   0.45,
P  A B   0.68 . Show that the events A and B are neither mutually exclusive nor
independent.
(ii) A bag contains 12 red balls, 8 blue balls and 4 white balls. Three balls are taken from
the bag at random and without replacement. Find the probability that all three balls are of
the same colors.
Find also the probability that all three balls are of different colors.
5. A box contains 25 apples, of which 20 are red and 5 are green. Of the red apples, 3
contains maggots and of the green apples, 1 contains maggots. Two apples are chosen at
random from the box. Find, in any order,
(i) the probability that both apples contain maggots,
(ii) the probability that both apples are red and at least one contains maggots,
(iii) the probability that at least one apple contains maggots, given that both apples are
red,
(iv) the probability that both apples are red given that at least one apple is red.
6. (a) Two digits X and Y are taken from a table of random sampling numbers. Event R is
that X  Y 1 and events S is that X and Y are both less than 2 . Write down
(i) P  R 
(ii) P  RS
(iii) P  RS
(iv) P  R | S 
(b) Conveyor belting for use in mines is tested for both strength and safety (the safety test
is based on the amount of heat generated if the belt snaps). A testing station receives belting
from three different suppliers : 30% of its tests are carried out on samples of belting from
supplier A , 50% from B ; 20 % from C . From past experience the probability of failing the
strength test is 0.02 for a sample from A, 0.12 from B and 0.04 from C .
(i) What is the probability that a particular strength test will result in a failure?
(ii) If a strength test result in a failure, What is the probability that the belting
came from supplier A?
(iii)What is the probability of a sample failing the safety tests given the following
further information:
supplier A - the probability of failing the safety tests is 0.05 and is independent of the
probability of failing the strength test;
supplier B – 1% the probability of samples fail both strength and safety test
supplier C – exactly half the samples which fail the strength test also fail the safety test
Common errors
1. Single probability
If there are n outcomes to an experiment, then each has probability only if they are
equally likely
When two dice are rolled, there are 11 possible for total score.
But a total of 12 is less likely than total of 7,so neither has probability
2. Addition of probability
The probability of 'A or 'B is only the sum of the probabilities if the events concerned
are exclusive. In general:
P  A or B   P  A   P  B 
3. Multiplication of probabilities
The probability of ‘A & ‘B is only the product of the probabilities if the events
concerned are independent.
4. Conditional probability
(a) Do not forgot to divide by P  B  when working out P  A | B 
(b) In the formula
P  A | B 
P( A & B)
P( B)
do not assume that P  A & B  is P  A   P  B  This is only true if A and B are
independent.
(c) Conditional probability is concerned with belief, not with cause and effect. If
P  A | B   P  A  . then it does not follow that B has caused A or prevented A . It may
even be that B happened after A did.
Solution (to exercise)
20.1.2
1
1. (a)
36
1
(e)
6
1
2. (a)
4
5
3. (a)
18
20
4. (a)
171
1
5. (a)
36
1
6. (a)
9
1
7. (a)
6
1
8. (a)
80
1
9. (a)
32
1
10. (a)
81
32
11. (a)
243
20.2.2
1. A & C , A & D, B & D
(b)
1
36
11
36
1
(b)
17
(c)
1
6
(d)
5
12
(f)
(c)
(b)
4
9
31
57
11
(b)
36
4
(b)
9
5
(b)
36
567
(b)
6400
31
(b)
32
65
(b)
81
211
(b)
243
(b)
(c)
91
171
(c)
25
216
2. A & C , A & D, B & D indep. B & C , C & D excl.
4
663
4
(d)
9
(d)
(c)
indep. None exc.
3. A & B indep. C & D exclusive.
4. A & D indep. B & C excl.
4
13
5. (a)
(b)
51
51
1
1
6. (a)
(b)
5
6
1
7. (a)
(b) 0
5
13
204
5
6
(d)
125
216
8. (a)
9.
1
4
(b)
3
4
6
10
35
68
35
11. (a)
68
12. (a) 0.9712
7
13.
8
4
14.
9
5
15.
6
1
16. 0.1,
2
17. 0.15, 0.55, 0.375
5
8
5
(b)
8
(b) 0.3125
10. (a)
(b)
18. (a) 0.8, 0
(b) 0.65, 0.15
1
2
1
(c)
2
(c) 0.999
(c)
===========================================================
References:
Solomon, R.C. (1997), A Level: Mathematics (4th Edition) , Great Britain, Hillman
Printers(Frome) Ltd.
More: (in Thai)
http://home.kku.ac.th/wattou/service/m456/10.pdf