**Probability Exercise**

*Note*: parts 1 and 2 of the exercise can be done separately. The results are given in the form of fractions.

16 passengers booked tickets in Terminal A so that:

7 of them go to station B (50 dollars per ticket).

5 of them go to station C (60 dollars per ticket).

4 of them go to Terminal D (75 dollars per ticket).

## 1. We randomly choose one of these travelers.

Let X be the random variable that attaches to each passenger the price of his ticket in dollars.

### a) Determine the probability law for the random variable X

### b) Calculate the expected value of the random variable X.

## 2 . We randomly choose three of these travelers.

### a) Calculate the probability that these travelers will have different directions.

### b) Calculate the probability that at least one passenger will be heading towards station B.

### c) What is the probability that the direction of the three passengers will be station B, given that they are traveling in the same direction.

**The solution**

**1.** We have X is the random variable that attaches to each passenger the price of their ticket in dollars.

### a) Determining the probability law to the random variable X.

7 passengers out of 16 are heading to Terminal B, where the ticket price is 50 dollars.

so: P(x=50)=7/16

5 out of 16 passengers are heading to Terminal C, where the ticket price is 60 dollars.

so: P(X=60)=5/16

4 out of 16 passengers are heading to Terminal D, where the ticket price is 75 dollars.

so: P(X=75)=4/16=1/4

Hence the probability law for variable X is:

### b) Calculate the expected value of the random variable X.

The expected value is given by the relation:

E(X)= 50.p(X=50)+60.P(X=60)+75.P(X=75)=475/8

So the expected value for a random variable X is 475/8=59.375

2. We randomly choose three of these travelers.

### a) Calculate the probability that these travelers will have different directions.

### b) Calculate the probability that at least one passenger direction will be station B.

So the probability that at least one passenger will be heading towards Terminal B is: 17/20.

### c) Calculate the probability that the direction of the three passengers will be station B, given that they are traveling in the same direction.

* Secondly, we calculate the probability p(F) of event F: “The direction of the three passengers will be station B.”

agree the conditional probability PE(F). By arithmetic, we find: