Question

Let X represent the outcome when a fair six-sided die is rolled. For this random variable, μX=$3.5$and σX=$1.71$. If the die is rolled 100 times, what is the approximate probability that the sum is at least $375$?

a. $0.0000$

b.$0.0017$

c.$0.0721$

d.$0.4420$

e.$0.9279$

Short answer

The probability is $0.0721$

Step-by-step solution

Given Information

We have to find the probability that sum is at least $375.$

Simplification

Population mean $\left(\mu x\right)=3.5$

Population standard deviation$\left(\sigma x\right)=1.71$

Sample size $(n)=100$

The mean can be calculated as follows:

$\stackrel{-}{x}=\frac{375}{100}=3.75$

The likelihood that the sum is at least $375$can be computed as follows:

$$\begin{gathered} P(\stackrel{-}{x}\ge 3.75)=P\left(\frac{x-\mu }{{\displaystyle \frac{\sigma }{\sqrt{n}}}}\ge \frac{3.75-3.5}{{\displaystyle \frac{1.71}{\sqrt{100}}}}\right) \\ =P(Z\ge 1.47) \\ =1-P(Z\le 1.47) \\ =0.0721 \end{gathered}$$

Thus, the required probability is$0.0721.$