One thousand independent rolls of a fair die will be made. Compute an approximation to...

Betsy Rhone

Betsy Rhone

Answered

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.

Answer & Explanation

Carl Swisher

Carl Swisher

Expert

2021-12-24Added 28 answers

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[ess]=P[6]=16
Let X be the random variable for the number of sixes in the 1000 rolls of the die. Then,
XB(1000,16)
Since, n is very large, the binomial random variable can be approximated as nominal random variable with mean μ=np=1000×16=166.67 and variance σ2=np(1p)=1000×16×56=138.85. Thus
XN(166.67,138.85)
Now, [150X200=P[150166.6711.78Xμσ200166.6711.78]
=P[Z<0.928]
=Φ(0.928)
=0.1767

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