Betsy Rhone

2021-12-23

One thousand independent rolls of a fair die will be made. Compute an approximation to the probability that the number 6 will appear between 150 and 200 times inclusively. If the number 6 appears exactly 200 times, find the probability that the number 5 will appear less than 150 times.

Carl Swisher

Expert

Given number of rolls of the die are $n=1000$. Let the event of six coming up be succes. Then, in each trial, the probability of success is
$p=P\left[\succ ess\right]=P\left[6\right]=\frac{1}{6}$
Let X be the random variable for the number of sixes in the 1000 rolls of the die. Then,
$X\sim B\in \left(1000,\frac{1}{6}\right)$
Since, n is very large, the binomial random variable can be approximated as nominal random variable with mean $\mu =np=1000×\frac{1}{6}=166.67$ and variance ${\sigma }^{2}=np\left(1-p\right)=1000×\frac{1}{6}×\frac{5}{6}=138.85$. Thus
$X\sim N\left(166.67,138.85\right)$
Now, $\left[150\le X\le 200=P\left[\frac{150-166.67}{11.78}\le \frac{X-\mu }{\sigma }\le \frac{200-166.67}{11.78}\right]$
$=P\left[Z<-0.928\right]$
$=\mathrm{\Phi }\left(-0.928\right)$
$=0.1767$

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