Question

We roll two fair 6-sided dice. Each one of the 36 possible outcomes is assumed to be equally likely. (a) Find the probability that doubles were rolled. (i.e. the two outcomes are equal) (b) Given that the roll resulted in a sum of 6 or less, find the conditional probability that doubles were rolled. (c) Find the probability that at least one die is a 1

Answer 1

Answer:

a) There is a 16.6% probability that doubles were rolled.

b) Given that the roll resulted in a sum of 6 or less, there is a 20% probability that doubles were rolled.

c) There is a 30.55% probability that at least one die is a 1.

Step-by-step explanation:

The probability formula is the number of desired outcomes divided by the number of total outcomes.

(a) Find the probability that doubles were rolled. (i.e. the two outcomes are equal)

There are six desired outcomes: (1,1), (2,2), (3,3), (4,4), (5,5), (6,6)

There are 36 total outcomes. So:

$P = \frac{6}{36} = 0.166$

There is a 16.6% probability that doubles were rolled.

(b) Given that the roll resulted in a sum of 6 or less, find the conditional probability that doubles were rolled.

The following rolls result in a sum of 5 or less:

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

So, there are 15 total outcomes.

3 of them are doubled: (1,1), (2,2), (3,3). So

$P = \frac{3}{15} = 0.2$

Given that the roll resulted in a sum of 6 or less, there is a 20% probability that doubles were rolled.

(c) Find the probability that at least one die is a 1

The following outcomes have at least one die that is 1:

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

So, there are 11 desired outcomes out of 36.

$P = \frac{11}{36} = 0.3055$

There is a 30.55% probability that at least one die is a 1.