Question

A die is rolled 12 times. Find the probability of rolling exactly 1 six.
The probability is:

Answer 1

The probability of rolling exactly one six in 12 rolls of a die is calculated using the binomial probability formula, resulting in
`P(X=1) = 12 * (1/6) * (5/6)^11.`

Step-by-step explanation:

The probability of rolling exactly one six in 12 rolls of a fair six-sided die can be calculated using the binomial probability formula. The probability of rolling a six on any given roll is 1/6, and the probability of not rolling a six is 5/6. We want to calculate the probability of getting exactly one six in 12 rolls, which means 11 rolls do not result in a six. The formula for the binomial probability P(X=k) where 'k' is the number of successes (rolling a six in this case), 'n' is the number of trials (12 rolls), and 'p' is the probability of success on any given trial (1/6 for rolling a six) is given by:

`P(X=k) = C(n,k) * p^k * (1-p)^(n-k)`

Here, C(n,k) is the combination of n objects taken k at a time and is calculated by:

`C(n,k) = n! / (k! * (n-k)!)`

So for P(X=1), where X is the random variable representing the number of sixes rolled:

`P(X=1) = C(12,1) * (1/6)^1 * (5/6)^(11)`

Calculating this:

`P(X=1) = 12 * (1/6) * (5/6)^11`

This calculation provides the probability of rolling exactly one six in 12 rolls of a die.