Download Probability_of_Compound_events

PROBABILITY OF COMPOUND
EVENTS
Independent Events
Independent events are
events that do not affect
each other
• If two events, A and B, are independent, then
the probability of both events occurring is the
product of each individual probability.
P AandB  P( A)  P( B)
According to the U.S. DOT statistics, the top ten airlines in
the U.S. arrive on time 80% of the time. If you have a
vacation planned that requires you to take four flights,
each on a separate day, what is the probability that all of
your flights will arrive on time?
(0.8)  (0.8)  (0.8)  (0.8)  .4096
Try these…
1. Using a standard deck of cards, find the probability of selecting
a face card, replacing it in the deck, and then selecting an ace.
12 4
3


 .01775
52 52 169
2. Find the probability of rolling a sum of 7 on the first toss of two
number cubes and a sum of 4 on the second toss.
6 3
1


 .01389
36 36 72
3. Given a bag of marbles containing 5 red, 4 black, and 7 yellow,
find the probability of selecting a red marble, replacing it in the
bag, selecting a yellow marble, and after replacing it selecting a
black marble.
5 7 4
35


16 16 16

1024
 .03418
4. Using a standard deck of cards, find the probability of selecting
a spade, replacing it in the deck, and then selecting a face card.
13 12 3


 .05769
52 52 52
and these…
5. The probability that a coal miner in a particular region will
develop black lung disease is 5/11. The probability that a
miner will develop arthritis is 1/5. If one health problem does
not affect the other, what is the probability that a randomlyselected miner will not develop black lung but will develop
arthritis?
6 1 6

11 5

55
 .10909
6. Of our first 42 American presidents, 4 were members of the
Whig party and 7 were born in Ohio.
a. Find the probability that a president selected at random
from this group was born in Ohio and from the Whig party.
7 4
1


 .01587
42 42 63
b. Find the probability that a president selected at random
from this group was born in Ohio but not from the Whig
party.
7 38 19

42 42

126
 .15079
Dependent Events
Independent events are
events that DO affect each
other
• If two events, A and B, are dependent, then
the probability of both events occurring is the
product of each individual probability.
P AandB  P( A)  P( BfollowingA)
Suppose you draw a card from the deck and then,
without replacing the card, draw a second card.
What is the probability of selecting two face cards?
12 11 11
 
 .04977
52 51 221
Try these…
1. Sam has 3 rock, 4 country, and 2 jazz CDs .
What is the probability that Sam will pull 2 rock
CDs if pulled without looking? 3  2  1  .08333
9 8
12
2. I have a bag of 4 red, 6 green, and 3 brown
M&M’s. If you choose 3, what is the probability
that all 3 will be brown? 3 2 1
1


13 12 11

286
 .00350
3. A drawer contains 6 black and 4 navy pairs of
socks. What is the probability that Ned will
randomly select two pairs of navy socks?
4 3 2
 
 .13333
10 9 15
4. A drawer contains 6 black and 4 navy pairs of
socks. What is the probability of selecting a
6 4 4
black pair of socks and
 
 .26667
10 9 15
a navy pair of socks?
Determine whether independent or
dependent. Then determine the probability.
7
 .19444
36
1. What is the probability of selecting a blue marble, not replacing
it, and then a yellow marble from a box of 7 blue and 2 yellow?
2.
1
 .11111
9
1
36
What is the probability of selecting two oranges from a bowl of 3
oranges and 6 tangerines, if the first selection is replaced?
3. A green number cube and a red number cube are tossed. What
is the probability that a 2 is shown on the green cube and a 3 is
 .02778 shown on the red cube?
4.
7
 .31818
22
10
 .03861
259
What is the probability of randomly taking 2 blue notebooks
from a shelf containing 7 blue and 5 black notebooks?
5. A piggy bank contains 10 nickels, 15 dimes, and 12 quarters.
What is the probability of selecting one nickel, then one dime,
and lastly one quarter without replacement?
6. In number 5 above, does the probability change if the question
specified selecting a quarter, then a dime, and lastly a nickel?
10
 .03861
259
D
No
I
I
D
D
D
Mutually Exclusive
Events
Mutually exclusive events cannot happen at the same time
• If two events, A and B, are mutually exclusive,
then the probability that either A or B occurs
is the sum of their individual probabilities.
P AorB  P( A)  P( B)
Suppose you have 8 nickels, 5 dimes, and 4
quarters in your pocket. If you pull out only one
coin, what is the probability that is will be a dime or
a quarter?
5
4
9


 .52941
17 17 17
Try these…
1. A bag of marbles contains 5 red, 4 black, 3 green
and 6 yellow. What is the probability of selecting
either a red or yellow marble?
5 6 11
 
 .61111
18 18 18
2. What is the probability of choosing a King or a
Queen from a standard deck of cards?
4
4
2


 .15385
52 52 13
3. What is the probability of two number cubes being
tossed and showing a sum of 6 or a sum of 9?
5
4 1

  .25
36 36 4
4. A sock drawer contains 5 black pairs, 4 navy pairs,
and 6 khaki pairs of socks. What is the probability
of randomly selecting either black or khaki?
5 6 11
 
 .73333
15 15 15
Inclusive Events
Inclusive events - can occur at
the same time
• If two events, A and B, are inclusive, then the
probability that either A or B occurs is the sum
of their individual probabilities decreased by the
probability of both occurring.
P AorB  P( A)  P( B)  P( AandB)
Consider the chart showing the outcomes of tossing
two die. What is the probability of rolling two
number cubes, in which the first cube shows a 2 or
the sum of the cubes is 6 or 7?
P(2)  P(6or 7)  P(2and 6or 7)
6 11 2
5



 .41667
36 36 36 12
Try these…
1. What is the probability of selecting a boy or a blondehaired person from 12 girls, 5 of whom have blonde hair,
and 15 boys, 6 of whom have blonde hair?
15 11 6 20



 .74074
27 27 27 27
2. In a particular group of hospital patients, the probability
of having high blood pressure is 3/8, the probability of
having arteriosclerosis is 5/12, and the probability of
having both is ¼. What is the probability that a patient in
this group has either high blood pressure or
3 5 1 13
arteriosclerosis?
  
 .54167
8 12 4 24
3. Suppose the probabilities for a driver’s license applicant
to pass the road test the first time is 5/6 and the written
exam on first attempt is 9/10. If the probability of passing
both on first attempts is 4/5, what is the probability that
an applicant would pass either examination on first
5 9 4 14
attempt?
  
 .93333
6 10
5
15
… Combinations
There are 5 students and 4 teachers on the school
newspaper committee. A group of 5 members is
being selected at random to attend a workshop.
What is the probability that the group attending
will have at least 3 students?
P(≥3 students) = P(3s & 2t) + P(4s & 1t) + P(5 s & 0t)
C3 4 C2 5 C4 4 C1 5 C5 4 C0



9 C5
9 C5
9 C5
60 20
1



126 126 126
9

 .64286
14
5
… Combinations
There are 4 girls and 2 boys in an art class. Four
will be chosen at random to act as greeters for the
school’s art exhibit. What is the probability that at
least three girls will be chosen?
P(≥3 girls) = P(3g & 1b) + P(4g & 0b)
C3 2 C1 4 C4 2 C0


6 C4
6 C4
8 1
 
15 15
3
  .6
5
4
… Combinations
1. The probability of randomly picking 5 puppies
of which at least 3 are male, from group of 5
males and 4 females?
C3 4 C2 5 C4 4 C1 5 C5 4 C0 9




 .64286
14
9 C5
9 C5
9 C5
5
2. The probability that a group of 6 people selected
from 7 men and 7 women will have at least 3
women?
C3 7 C3 7 C4 7 C2 7 C5 7 C1 7 C6 7 C0 302





 .70396
429
14 C6
14 C6
14 C6
14 C6
7
Mixed practice…
Find the probability of each using a standard deck of cards.
1. P(all red cards) if 5 cards are drawn without replacement.
26 25 24 23 22 253
   

 .02531
52 51 50 49 48 9996
2. P(both kings or both aces) if 2 cards drawn without replacement.
2
 4 3  4 3
 .00905
    
 52 51   52 51  221
3. P(all diamonds) if 10 cards selected with replacement.
10
1
 13 
 .00000095
  
1,048,576
 52 
4. P(both red or both queens) if 2 drawn without replacement.
 26 25   4 3   2 1  55
 .24887
       
 52 51   52 51   52 51  221
Mixed practice…
There are 5 pennies, 7 nickels, and 9 dimes. Two coins are
selected without replacement. Find each probability.
1. P(2 pennies).
5 4
1


 .04762
21 20 21
2. P(2 nickels or 2 silver-colored coins).
 7 6   16 15   7 6  4
             .57143
 21 20   21 20   21 20  7
3. P(at least 1 nickel)
 14 13  17
1    
 .56667
 21 20  30
4. P(2 dimes or 1 penny and 1 nickel).
 9 8   5 7  107
 .25476
    
 21 20   21 20  420
Mixed practice…
There are 5 male and 5 female students in the executive council of
the Honor Society. A committee of 4 members is to be selected at
random to attend a conference. Find each probability.
1. P(all female).
5
C4 5 C0
1

 .02381
42
10 C4
2. P(all female or all male).
5
C4 5 C0 5 C0 5 C4
1
1
1




 .04762
42 42 21
10 C4
10 C4
3. P(at least 3 females)
5
C3 5 C1 5 C4 5 C0 11


 .26190
42
10 C4
10 C4