Answer to Question #340122 in Statistics and Probability for Maneo

Question

If two dice are rolled, find the probability that:

• A. both show an odd number

• B. a sum of 7 shows

• C. a sum of 4 or 11 shows

• D. only the second die shows a 3.

Answer 1

Probability can be defined as the ratio of the number of favorable outcomes to the total number of outcomes of an event.

• A For an experiment having n number of outcomes, the number of favorable outcomes can be denoted by:

Let ‘x’ be the number on the first dice

‘Y’ be the number on second dice

First dice showing odd number = {1,3,5}

Second dice also has odd number = {1,3,5}

The probability that the first dice shows an odd number = 3/6.

The probability that the second dice shows an odd number = 3/6.

The possible results are (1,1), (1,3), (1,5), (3,1), (3,3), (3,5), (5,1), (5,3), (5,5).

The probability that both dice show an odd number is (3/6) × (3/6)

= 9/36 = 1/4

Therefore, the probability of getting an odd number in both dice is 1/4.

• B When two dice are rolled together then total outcomes are $6^2=36$.

Pairs with sum 7 are (1, 6) (2, 5) (3, 4) (4, 3) (5, 2) (6, 1) i.e. total 6 pairs

Total outcomes = 36

Favorable outcomes = 6

Probability of getting the sum of 7 = Favorable outcomes / Total outcomes

                           = 6 / 36 = 1/6

So, P(sum of 7) = 1/6.

• C Total number of outcomes is 36. Consider the possible events that lead to a sum of 4. We have 1+3, 2+2, 3+1 (Total 3). Consider the possible events that lead to a sum of 11. We have 5+6, 6+5 (Total 2).

So P(sum of 4) = 3/36 and P(sum of 11) = 2/36

So, P(sum of 4 or sum of 11) = P(sum of 4) + P(sum of 11) = 3/36 + 2/36 = 5/36.

• D  The chance of getting a 3 on one die is 1/6. The chance of not getting a 3 on one die is 5/6. Thus the probability of getting a 3 on exactly one die is 1/6 * 5/6 = 5/36.