Question

A fair die is rolled five times. A 5 is considered success while all other outcomes are failures. Find the probability of 0 successes out of 5 rolls.

Answer 1

SOLUTION

When a probability deal with obtaining succes a number of times, it is Binomial probability distribution.

The formula for Bimomial probability distribution is

$\begin{gathered} p(x)=^nC_xp^xq^{n-x} \\ \text{where } \\ n=\text{ Number of trials} \\ p=probability\text{ of success} \\ q=1-p \\ x=total\text{ number of successes} \end{gathered}$

For the given question, we have

$\begin{gathered} n=5,x=0 \\ p=\frac{1}{6},q=1-\frac{1}{6}=\frac{5}{6} \end{gathered}$

A die has six total outcome, the probability of obtaining a number out of six is 1/6, this account for the value of p above.

stituting the value into the formula, we have

$\begin{gathered} p(x)=^5C_0(\frac{1}{6})^0(\frac{5}{6})^5 \\ \text{Recall that }^5C_0=1,\text{ then} \\ p(x)=1\times1\times(\frac{5}{6})^5 \\ P(x)=\frac{3125}{7776}=0.4019 \end{gathered}$

Therefore

The probability of 0 successes out of 5 rolls when a die is rolled five times and a 5 is considered success is 0.4019.

Answer = 0.4019