Question

Rolling a Die. If we repeatedly roll a balanced die, then, in the long run, it will come up "4" about one-sixth of the time. But what is the probability that such a die will come up "4" exactly once in six rolls?

Short answer

The probability that "4" come up exactly once in six rolls is 0.246.

Step-by-step solution

Step 1. Given information.

The statement given in the question is:

If we repeatedly roll a balanced die, then, in the long run, it will come up "4" about one-sixth of the time.

Step 2. Find the probability.

When we roll a balanced die, it will come up "4" around one-sixth of the time in the long run.

When we roll a die, the possible outcomes are 1, 2, 3, 4, 5, and 6. The chance of getting event 4 is $\frac{1}{6}$.

Therefore,$n=6\mathrm{and}p=\frac{1}{6}$.

Step 3. Find the probability.

Let Xbe the total number of successful Bernoulli trials with probability in n Bernoulli trials:

$$\begin{gathered} P\left(X=x\right)=\left(\begin{array}{c}n\\ x\end{array}\right)p^x{\left(1-p\right)}^{n-x} \\ P\left(X=1\right)=\left(\begin{array}{c}6\\ 1\end{array}\right){\left(\frac{1}{6}\right)}^1{\left(1-\frac{1}{6}\right)}^{6-1} \\ =\frac{6!}{1!\left(6-1\right)!}{\left(\frac{1}{6}\right)}^1{\left(\frac{1}{6}\right)}^5 \\ =6\left(0.1667\right)\left(0.4018\right) \\ =0.402 \end{gathered}$$