Question 1

An experiment consists of picking a card at random from a standard deck of cards and replacing it. If this experiment is performed 12 times, what is the probability that you get:

(a) exactly 2 aces;

(b) exactly 3 hearts;

(c) more than 1 heart.

Solution

Let n = 12, the no of experiments.

(a) Let p = probability of obtaining an ace in an experiment of picking a card at random from a standard deck of cards; and = 4/52 = 1/13

let x = no of aces obtained in 12 experiments.

Pr(x = 2) = 12C2 (1/13)2(12/13)10

= 0.1754

(b) Let p = probability of obtaining a heart in an experiment of picking a card at random from a standard deck of cards; and = 13/52 = 1/4

let x = no of hearts obtained in 12 experiments.

Pr(x = 3) = 12C3 (1/4)3(3/4)9

= 0.2581

(c) With p and x defined from (b),

Pr(x > 1) = 1 – Pr(x = 0) – Pr(x = 1)

= 1 – 12C0 (1/4)0(3/4)12 – 12C1 (1/4)1(3/4)11 = 1 – 0.0317 – 0.1267

= 0.8416

Question 2

In a small town, there are two post offices. The number of customers arriving at each of the post office is a Poisson process, with an average of 6 customers per hour at post office A, and 4 customers per hour at post office B. The number of customers arrival at the two post offices are independent of one another.

(a) What is the probability that there are exactly 2 customers arrive at the two post offices in a period of 30 minutes?

(b) What is the probability that there are more than 2 customers arrive at the two post offices in a period of 30 minutes?

Solution

Let xA = number of customers arriving at post office A in 30 minutes with λA = 3;

xB = number of customers arriving at post office B in 30 minutes with λB = 2

(a) Pr(there are exactly 2 customers arrive at the two post offices in 30 minutes)

= Pr(xA + xB = 2)

= Pr(xA = 0,xB = 2) + Pr (xA = 1,xB = 1) + Pr(xA = 2,xB = 0)

= Pr(xA =