Compound Probabilities and Odds

-- Sections 8.8,8.9 --
 
Roll two dice at once.  
  • Find the probability of throwing a sum of 10.
  • Find the probability of throwing a sum of 2 or 3.
dice.wmf (2966 bytes) (1,1) (1,2) (1,3) (1,4) (1,5) (1,6)
(2,1) (2,2) (2,3) (2,4) (2,5) (2,6)
(3,1) (3,2) (3,3) (3,4) (3,5) (3,6)
(4,1) (4,2) (4,3) (4,4) (4,5) (4,6)
(5,1) (5,2) (5,3) (5,4) (5,5) (5,6)
(6,1) (6,2) (6,3) (6,4) (6,5) (6,6)
An experiment consists of throwing two dice, one red and one green.  The reason for using 2 different color dice  is to emphasize that (1,2)  and (2,1) are 2 different throws.  

Solve

       Step
Find the probability of throwing a sum of 10.

You can roll   (4,6) , (5,5), or  (6,4)

3 ways to roll a 10.

6 × 6 = 36 different throws.

Find the probability of throwing a sum of 2 or 4.

You can roll   (1,1) , (1,3),  (2,2) or  (3,1)

 1 way to roll a 2 and 3 ways to roll a 4

4 ways total,  È means 'or' 

  
  Roll two dice at once.  
  • Find the probability of throwing a sum of 7.
  • Find the probability of throwing a sum of 7 or 11.

[Solution]

Suppose we pick 2 balls out of a box  without replacement. Let's say 5 are red and 7 are blue. What is the probability of picking a red ball then a blue ball?

Solve

        Step
Pick red first.  Pr(red) = 5/12 

There are a total of 12 balls in the box.

Then pick blue. Pr(blue) = 7/11

 Notice it is not 7/12 since once we picked a red ball there is one less ball in the box.

Pr(red then blue) = 5/12 × 7/11 = 35/121 

 These events are not independent.

 

  Suppose we pick 2 balls out of a box  without replacement. Let's say 3 are red and 5 are blue. What is the probability of picking a red ball then a blue ball?

[Solution]

Suppose we pick 2 balls out of a box  with replacement. Let's say 5 are red and 7 are blue. What is the probability of picking a red ball then a blue ball?
Solve

Step
Pick red first.  Pr(red) = 5/12 

There are a total of 12 balls in the box.

Then pick blue.  Pr(blue) = 7/12

 The number of balls is not reduced because the question said with replacement. The first ball is put back in the box.

Pr(red then blue) = 5/12 × 7/12 = 35/144 

 These events are independent.

Suppose we pick 2 balls out of a box  with replacement. Let's say 8 are red and 4 are blue. What is the probability of picking a red ball then a blue ball?

[Solution]

The odds of winning a race is 3 to 2.  Find the probability of winning.

If the odds in favor of an event are a to b,  the probability of the event is  

Solve

Step

a = 3 and b = 2

The probability of winning is 60%.  

The probability of winning a race is 0.70.  Find the odds in favor of winning.

[Solution]

Tutorials and Applets by
Joe McDonald
Community College of Southern Nevada