Die rolling

Question

Two standard six-faced die are rolled. Jean wins if the product of the two numbers is odd or a multiple of three, otherwise, Allen wins. What is the probability that jean wins?

Answer 1

To get an odd product requires that both of the dice are odd. This will occur 1/4 of the trials or 9 out of 36. For the product to be a multiple of 3 requires that either die is 3 or 6. this will occur 20 times out of 36. Thus it may appear that the answer is 29 out of 36, but it is not. You cannot count (3,3) (3,1) (1,3) (1,6) (6,1) twice, so subtract off 5 out of 36. The answer is 24 out of 36 or 2/3 for the probability that Jean wins. Allen wins only 1/3 of the time. One could say that Jean has a 2 to 1 advantage in this game.

Answer 2

Lets define event A as the product of the two numbers being odd, and lets define event B as the product of the two numbers being a multiple of 3.

 

First let's consider event A. we know that only an odd number times an odd number will produce an odd number (e.g. 1*3 = 3). An even number times an even number will produce an even number, and an odd number times an even number will also produce even. So for event A, we only consider two odd numbers. so the possibilities are:

 

(1,1) , (1,3), (1,5), (3,1), (3,3), (3,5), (5,1), (5,3), (5,5)

 

So we have 9 possibilities to satisfy the condition. But what is the total number of possibilities when you roll two standard six-faced die?  It is 6*6 = 36  

 

So the probability of the product being odd would be:   9/36

 

Now let's consider event B. Multiples of 3 are: 3,6,9,12,15,18,21,24,27,30,33,36

Let's see how we can obtain any of those numbers.

 

(1,3), (3,1), (3,2), (2,3), (3,3), (3,4), (4,3), (3,5), (5,3), (3,6), (6,3), (4,6), (6,4), (5,6), (6,5), (6,6)

 

Note that we can not obtain 21 or 33, and the highest we can get is 36 when we roll double 6s.

 

So the probability of event B would be 16/36

 

Now, we have to make sure that we have not double counted anything. So we will add the two probabilities, but also subtract any common factors.

There are 5 common factors between the two events: (1,3), (3,1), (3,3), (3,5), (5,3)

 

So the probability that Jean wins would be:

 

(9/36) + (16/36) - (5/36) = 20/36 = 5/9

 

Please note that I have used the formula P(A U B) = P(A) + P (B) - P (A intersection B)

Please let me know if you have any questions.