Question
Asked Jul 31, 2019

Two fair dice are rolled at once. Find the conditional probability that the sum is 6, given that the sum is an even number.

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Step 1

Concept of conditional probability:

For any two events A and B with P(B) >0, the formula for conditional probability is given by:

P(A|B)- P(AnB)
P(B)
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P(A|B)- P(AnB) P(B)

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Step 2

Find the sample space:

Sample space:

The set of all possible outcomes of a probability experiment is called sample space of the experiment.

Here, two fair dice are rolled.

The total number of outcomes in the sample space be n(S) = 62 = 36. That is, there will be 36 equally likely outcomes.

Outcomes will be occurred in ordered pairs. First number in each ordered pair represents the number on the first die and second number in each ordered pair represents the number on the second die. Each of the two numbers can take values 1 to 6.

The sum of the numbers on the dice will be 2, 3, 4, 5, 6, 7, 8, 9, 10,11 and 12.

The sample space for the experiment is given below:

The possible outcomes for the sum to be 2:{(1,1)}
The possible outcomes for the sum to be 3:{(1,2),(2,1)
The possible outcomes for the sum to be 4:{(1,3).(2,2),(3,1
The possible outcomes for the sum to be 5:{1,4),(2,3).(3,2).(4,1
The possible outcomes for the sum to be 6: {(1,5),(2,4),(3,3).(4,2), (5,1)}
The possible outcomes for the sum to be 7:{(1,6).(2,5).(3,4).(4,3). (5,2).(6,1)
The possible outcomes for the sum to be 8:{(2,6).(3,5), (4,4),(5,3).(6,2)}
The possible outcomes for the sum to be 9:{(3,6),(4,5).(5,4),(6,3)
The possible outcomes for the sum to be 10:{(4,6),(5,5),(6,4)
The possible outcomes for the sum to be 11:{(5,6).(6,5)}
The possible outcomes for the sum to be 12:{(6,6)
(1,1).(1,2), (1,3).(1.4).(1.5).(1,6).(2,1).(2,2).(2,3).(2,4).(2,5).(2,6),
Sample space S (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)
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The possible outcomes for the sum to be 2:{(1,1)} The possible outcomes for the sum to be 3:{(1,2),(2,1) The possible outcomes for the sum to be 4:{(1,3).(2,2),(3,1 The possible outcomes for the sum to be 5:{1,4),(2,3).(3,2).(4,1 The possible outcomes for the sum to be 6: {(1,5),(2,4),(3,3).(4,2), (5,1)} The possible outcomes for the sum to be 7:{(1,6).(2,5).(3,4).(4,3). (5,2).(6,1) The possible outcomes for the sum to be 8:{(2,6).(3,5), (4,4),(5,3).(6,2)} The possible outcomes for the sum to be 9:{(3,6),(4,5).(5,4),(6,3) The possible outcomes for the sum to be 10:{(4,6),(5,5),(6,4) The possible outcomes for the sum to be 11:{(5,6).(6,5)} The possible outcomes for the sum to be 12:{(6,6) (1,1).(1,2), (1,3).(1.4).(1.5).(1,6).(2,1).(2,2).(2,3).(2,4).(2,5).(2,6), Sample space S (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)

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Step 3

Find the conditional probability that the sum is 6, given that the sum is an even number:

There are 5 different ways the sum of the two dice is 6 and 18 different ways the sum of the two dice is an ...

P(sum is 6sum is an even number)
P(sum is an even number
P(sum is 6 sum is an even number)=
P(sum is 6)
P(sum is an even number)
5
36
18
36
5
18
0.27778
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P(sum is 6sum is an even number) P(sum is an even number P(sum is 6 sum is an even number)= P(sum is 6) P(sum is an even number) 5 36 18 36 5 18 0.27778

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