Hi, there--
THE PROBLEM:
A man tosses two fair dices. What is the conditional probability that the sum of the two dice will be 7, given that:
1) the sum is odd.
2) the sum is greater than 6.
3) the two dice had the same outcome?
Find P(A|B), P(A|C), P(A|D).
A SOLUTION:
1) Find the conditional probability that the sum of two dice is 7 given that their sum is odd.
Define the events you are interested in:
Let O be the event of rolling an odd sum.
Let S be the event that the sum of the dice is seven.
You want to find P(S|O). By definition, P(S|O)=P(SO)/P(O).
Calculate P(O):
There are 36 ways to roll two dice. The possible sums from die#1 and die#2 (D1,D2) are given
to the table below. (Note that order matters. A sum of 1+2 is a different outcome than 2+1.)
1 2 3 4 5 6
----------------
1| 2 3 4 5 6 7
2| 3 4 5 6 7 8
3| 4 5 6 7 8 9
4| 5 6 7 8 9 10
5| 6 7 8 9 10 11
6| 7 8 9 10 11 12
Counting, we see that there are 18 ways to achieve an odd sum:
(D1,D2) = {(1,2), (1,4), (1,6), (2,1), (2,3), (2,5),(3,2), (3,4),(3,6), (4,1), (4,3),(4,5), (5,2),(5,4), (5,6), (6,1),(6,3),(6,5)}
Therefore, P(O)=18/36.
Calculate P(SO):
Refer to the chart above. There are 6 ways to achieve a roll that is both odd and a sum of 7.
(You can verify that yourself.) Therefore, P(SO)=6/36.
Use the Conditional Probability Formula to calculate P(S|O):
P(S|O)=P(SO)/P(O) = (6/36)/(18/36) = 6/18 = 1/3
1) Find the conditional probability that the sum of two dice is 7 given that their sum is greater
than 6.
Define the events you are interested in:
Let S be the event that the sum of the dice is 7.
Let G be the event that the sum is greater than 6.
We want to find P(S|G). By definition, P(S|G)=P(SG)/P(G)
Calculate P(G):
Refer to the chart above. There are 21 ways to roll a sum greater than 6. Therefore,
P(G) = 21/36.
Calculate P(SG):
Refer to the chart again. The are 12 ways to roll a sum that is both odd and greater than 6.
Therefore, P(SG) = 12/36.
Calculate P(S|G):
P(S|G) = P(SG)/P(G) = (12/36)/21/36) = 12/21 = 4/7
3) Find the conditional probability that the sum of two dice is 7 given that the two dice had
the same outcome.
If we think about this for a minute, we see that the probability must be 0. There is no way to
roll a 7 when the dice have the same outcomes (i.e.rolling doubles.) Let's work through the
formula to prove that this is the case.
Define events:
Let S be the outcome that the sum of the dice is 7.
Let D be the outcome that both dice have the same outcome (doubles).
We want to find P(S|D).
Calculate P(D):
Refer to our trusty chart again. There are 6 ways for both dice to have the same outcome,
Therefore, the P(D) = 6/36
Calculate P(SD):
There are no ways for the dice to have the same outcome and have a sum of 7. So P(SD)=0.
Calculate P(S|D):
P(S|D) = P(SD)/P(D) = 0/(6/36) = 0
It's not clear from your from what P(A|B), P(A|B), and P(A|C) refer to because you have not
defined the events A, B, C, and D.
Hope this helps!
Mrs. Figgy
math.in.the.vortex@gmail.com