# Find the probability that 6 successes occurs in 10 rolls. What is the probability that the first...

## Question:

Find the probability that 6 successes occurs in 10 rolls.

What is the probability that the first success is on the 5th roll 5,

An experiment consists of rolling a die several times. Define success as obtaining a 2 or 3' in a roll.

A) Find the probability that 6 successes occur in 10 rolls

B) What is the probability that the first success occur on 5th roll?

## Binomial Experiment

A binomial experiment is a situation where there are an exact number of trials, the probability for success in each of the trials remains constant, and the trials are independent of each other. Examples of binomial experiments are rolling a die a fixed number of times with a certain number defined as success and flipping a coin a fixed number of times where heads or tails is defined as success.

## Answer and Explanation:

First, if we consider success to be getting a 2 or a 3 on a roll, then the probability of success in one roll is 2/6 or 1/3.

A) Consider first getting six rolls of 2 or 3 and then something else on the remaining 4 rolls. The probability of this is {eq}(1/3)^6*(2/3)^4 {/eq}. However, there are different ways to arrange the six success, i.e. a combination of ten items taken 6 at a time. Thus, the probability for getting any six rolls as success is {eq}10C6*(1/3)^6*(2/3)^4 = 0.0569 {/eq} rounded to four decimal places.

B) Here we really do not care about any roll after the fifth roll. So the first four rolls are not successes and the fifth is. The probability is then calculated as {eq}(2/3)(2/3)(2/3)(2/3)(1/3) = 0.0658 {/eq} rounded to four decimal places.

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