Random digits are integers selected from among 0,1,2,3,4,5,6,7,8,9 one at a time in such a way that at each stage in the selection process the integer chosen is just as likely to be one digit as any other. In simulation experiments it is often necessary to generate a series of random digits.
In generating such a series, consider the number of trials needed to obtain the first zero.
(a) What’s the probability that exactly three trials are needed?
(b) What’s the probability that no more than three trials are needed?
(c) What’s the probability that no less than a hundred trials are needed?
(d) What’s the expected number of trials needed to obtain the first zero?
+42 In Statistics, the discrete Geometric Distribution is used when we want to determine the probability until first success when considering a discrete random variable, say X. The following equations are very useful when considering the geometric distribution
P(X=x)=(1−p)x−1pP(X=x)=(1−p)x−1p
E(X)=1pE(X)=1p
P(X
P(X>x)=(1−p)xP(X>x)=(1−p)x
Answer and Explanation:
Since there are ten numbers, then the probability of success is
p=110p=110
From the formulas in the context, we can clearly see that
a)
P(X=3)=(1−110)2110=0.081P(X=3)=(1−110)2110=0.081
b)
P(X<3)=1−[1−110]4=0.3439P(X<3)=1−[1−110]4=0.3439
c)
P(X>100)=[1−110]100=2.66×10−5P(X>100)=[1−110]100=2.66×10−5
d)
E(X)=1110=10
