Probability Exercise with Answer | Example 1

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.

   We have:    

                 

Let the event V be for “the three travellers in different directions,” then we have:

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

We consider event U as: "None of the three passengers are Heading to station B."

Then the complementary event is Ū: “At least one passenger is heading towards station B.”

We know that 9 passengers have a direction other than terminal B, and from there

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.

* We first calculate the probability P(E) of event E: "The three passengers have the same direction".

This means station B, station C, or station D. These are mutually exclusive events, and from it:

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

The event "The probability that the direction of the three passengers is station B, Note that they are traveling in the same direction".

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

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