The binomial and geometric distributions

Find the probability that each of the following events occur.

a.  Exactly five heads are obtained when a fair coin is tossed nine times.

b.  Exactly two 6s are obtained with 11 rolls of a fair dice.

a. P(five heads) = `(9/5)(1/2)^5(1/2)^4 = 0.246`

b. P(two sixes) = `(11/2)(1/6)^2(5/6)^9 = 0.296`

A man has five packets and each contains three brown sugar cubes and one white sugar cube. He randomly selects one cube from each packet. Find the probability that he selects exactly one brown sugar cube.

P(one brown) = `(5/1)(3/4)^1(1/4)^4 = 0.0146`

Research shows that the owners of 63% of all saloon cars are male. Find the probability that exactly 20 out of 30 randomly selected saloon cars are owned by:

a. males

b. females.

a. Using p = 0.63 and q = 0.37 

P(20 male owners) = `(30/20) * 0.63^20*0.37^10 = 0.14`

b. Using p = 0.37 and q = 0.63 

P(20 female owners) = `(30/20)*0.37^20*0.63^10 = 0.000684`

In a particular country, 90% of both females and males drink tea. Of those who drink tea, 40% of the females and 60% of the males drink it with sugar. Find the probability that in a random selection of two females and two males:

a.  all four people drink tea

b.  an equal number of females and males drink tea with sugar

a. `0.9^4 = 0.6561`

b. 

P(2M and 2F) + P(1M and 1F) + P(0M and 0F) 

`(0.54^2*0.36^2)+(2*0.36*0.54)(2*0.54*0.36) + (0.36^2*0.54^2)= 0.227`

Given that `X~B(8,0.25),`calculate:

a. E(X) and Var(X)

b. P[X = E(X)]

c. P[X < E(X)] 

a. E(X) = `8*0.25`; Var(X) = `8*0.25*0.75 = 1.5`

b. P[X= E(X)] = P(X = 2) = `(8/2)*0.25^2*0.75^6 = 0.311`

c. P[X < E(X)] = P(X<2) = P(X = 0) + P(X =1) = `0.75^8 + (8/1)*0.25^1 *0.75^7 = 0.367`