Question

A box contains four 75 W lightbulbs, three 60 W lightbulbs, and three burned-out lightbulbs. Two bulbs are selected at random from the box without replacement. Let X represent the number of 75 W bulbs selected. Find the probability mass function for X. Show that X follows a valid probability mass function.

a. Find P( X > 0)
b. Find μx
c. Find σx^2

Answer 1

a. 0.689

b. 0.8

c. 0.427

Explanation:

The given scenario indicates hyper-geometric experiment because because successive trials are dependent and probability of success changes on each trial.

The probability mass function for hyper-geometric distribution is

P(X=x)=kCx(N-k)C(n-x)/NCn

where N=4+3+3=10

n=2

k=4

a.

P(X>0)=1-P(X=0)

The probability mass function for hyper-geometric distribution is

P(X=x)=kCx(N-k)C(n-x)/NCn

P(X=0)=4C0(6C2)/10C2=15/45=0.311

P(X>0)=1-P(X=0)=1-0.311=0.689

P(X>0)=0.689

b.

The mean of hyper-geometric distribution is

μx=nk/N

μx=2*4/10=8/10=0.8

c.

The variance of hyper-geometric distribution is

σx²=nk(N-k).(N-n)/N²(N-1)

σx²=2*4(10-4).(10-2)/10²*9

σx²=8*6*8/900=384/900=0.427