Question

roll two standard dice and add the numbers. what is the probability of getting a number larger than 8 for the first time on the third roll

Answer 1

Answer:

$Probability = \frac{845}{5832}$

Step-by-step explanation:

Given

Two standard dice

Required

Probability that the outcome will be greater than 8 for the first time on the third roll

First, we need to list out the sample space of both dice

$S_1 = \{1,2,3,4,5,6\}$

$S_2 = \{1,2,3,4,5,6\}$

Next, is to list out the sample when outcome of both dice are added together$S = \{2,3,4,5,6,7,3,4,5,6,7,8,4,5,6,7,8,9,5,6,7,8,9,10,6,7,8,9,10,11,7,8,9,10,11,12\}$

Next, is to get the probability that an outcome will be greater than 8

Represent this with P(E)

$P(E) = \frac{Number\ of\ outcomes\ greater\ than\ 8}{Total}$

$P(E) = \frac{10}{36}$

$P(E) = \frac{5}{18}$

Next, is to get the probability that an outcome will noy be greater than 8

Represent this with P(E')

$P(E) + P(E') = 1$

$P(E') = 1 - P(E)$

$P(E') = 1 - \frac{5}{18}$

$P(E') = \frac{18 - 5}{18}$

$P(E') = \frac{13}{18}$

Now, we can calculate the required probability;

Probability of a number greater than 8 first on the third attempt is:

Probability of outcome not greater than 8 on the first attempt * Probability of outcome not greater than 8 on the second attempt * Probability of outcome greater than 8 on the third attempt

Mathematically;

$Probability = P(E') * P(E') * P(E)$

Substitute values for P(E) and P(E')

$Probability = \frac{13}{18} * \frac{13}{18} * \frac{5}{18}$

$Probability = \frac{13 * 13 * 5}{18 * 18 * 18}$

$Probability = \frac{845}{5832}$