Conditional probability - Exercise Set 1

This exercise set contains some solved exercises on conditional probability. The theory needed to solve these exercises is introduced in the lecture entitled Conditional probability.

Exercise 1.1

Consider a sample space comprising three possible outcomes $omega _{1}$ , $omega _{2}$ , $omega _{3}$ :

Suppose the three possible outcomes are assigned the following probabilities: [eq2]

Define the events: [eq3]

and denote by $E^{c}$ the complement of .

Compute , the conditional probability of given $E^{c}$ .

Solution

We need to use the conditional probability formula: [eq5]

The numerator is: [eq6] and the denominator is:

As a consequence: [eq8]

Exercise 1.2

Consider a sample space comprising four possible outcomes $omega _{1}$ , $omega _{2}$ , $omega _{3}$ , $omega _{4}$ :

Suppose the four possible outcomes are assigned the following probabilities: [eq10]

Define two events: [eq11]

Compute , the conditional probability of given .

Solution

We need to use the formula: [eq13]

Butwhile, using additivity:

Therefore: [eq16]

Exercise 1.3

The Census Bureau has estimated the following survival probabilities for men:

probability that a man lives at least 70 years: 80%;
probability that a man lives at least 80 years: 50%.

What is the conditional probability that a man lives at least 80 years given that he has just celebrated his 70th birthday?

Solution

Given an hypothetical sample space , define the two events: [eq17]

We need to find the following conditional probability: [eq18]

The denominator is known:

As far as the numerator is concerned, note that (if you live at least 80 years then you also live at least 70 years). But implies:

Therefore:

Thus: [eq22]

by Marco Taboga

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