Question

Of the mountain climbers who attempt a particular peak, only 80% reach the summit.

If 15 climbers independently attempt the climb, find the probability that

A. at most 12 reach the summit

B. all 15 reach the summit given that more than 12 do

C. Use an approximating probability distribution to estimate the probability that if 100 climbers attempt the climb next year, at most 20 will fail to reach the summit.

Answer 1

here this is binomial with parameter n=15 and p=0.8

a)

P(X<=12)=

∑x=0a     (nCx)px(1−p)(n-x)    =

0.6020

if using ti-84 use commannd :binomcdf(15,0.8,12)

if using excel use commannd :binomdist(12,15,0.8,true)

B)

P(X=15|X>12) =P(X=15)/P(X>12) =0.8^15/(1-0.6020)=0.0884

C)

n=

100

p=

1-0.80 =0.2000

here mean of distribution=μ=np=		20.00
and standard deviation σ=sqrt(np(1-p))=		4.00
for normal distribution z score =(X-μ)/σx

therefore from normal approximation of binomial distribution and continuity correction:

probability =P(X<20.5)=(Z<(20.5-20)/4)=P(Z<0.13)=0.5517