Question

If you roll a die five times, what is the probability that you obtain exactly 2 fours?

Short answer

The probability of getting exactly two fours is$0.16$

Step-by-step solution

Step 1. Given information.

A die is rolled 5 times.

Die has 6 faces on which 6 different numbers: {1,2,3,4,5,6}.

Step 2. The probability of getting exactly 2 fours.

The probability of getting 4 in one roll is:$p=\frac{1}{6}$

The probability of not getting 4 in one roll is:

$$\begin{gathered} q=1-\frac{1}{6} \\ =\frac{5}{6} \end{gathered}$$

So, Probability of getting 2 fours in 5 times is:

$$\begin{gathered} P={}^{5}c_2{\left(p\right)}^2{\left(q\right)}^{5-2} \\ =\frac{5!}{2!(5-2)!}{\left(\frac{1}{6}\right)}^2{\left(\frac{5}{6}\right)}^3 \\ =\frac{5!}{2!3!}{\left(\frac{1}{6}\right)}^5.5^3 \\ =\frac{5\times 4\times 3\times 2\times 1}{2\times 1\times 3\times 2\times 1}\times 125{\left(\frac{1}{6}\right)}^5 \\ =\frac{1250}{6^5} \\ =\frac{1250}{7776} \\ =0.16 \end{gathered}$$