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STATLECT > Fundamentals of probability > Probability

Probability - Exercise set 1

This exercise set contains some solved exercises on sample spaces, probability and events. The theory needed to solve these exercises is introduced in the lecture entitled Probability.

Exercise 1.1

A ball is drawn at random from an urn containing colored balls. The balls can be either red or blue (no other colors are possible). The probability of drawing a blue ball is $1/3$. What is the probability of drawing a red ball?

nav_button Solution

The sample space Omega can be represented as the union of two disjoint events E and F:[eq1]where the event E can be described as 'a red ball is drawn' and the event F can be described as 'a blue ball is drawn'. Note that E is the complement of F:[eq2]

We know [eq3], the probability of drawing a a blue ball:[eq4]

We need to find [eq5], the probability of drawing a a red ball. Using the formula for the probability of a complement:[eq6]

Exercise 1.2

Consider a sample space Omega comprising three possible outcomes:[eq7]

Suppose the probabilities assigned to the three possible outcomes are:[eq8]

Can you find an event whose probability is $3/4$?

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There are two events whose probability is $3/4$.

The first one is:[eq9]

Using the formula for the probability of a union of disjoint events:[eq10]

The second one is:[eq11]

Using the formula for the probability of a union of disjoint events:[eq12]

Exercise 1.3

Consider a sample space Omega comprising four possible outcomes:[eq13]

Consider the three events E, F and $G$ defined as follows:[eq14]

Suppose their probabilities are:[eq15]

Now, consider a fourth event H defined as follows:[eq16]

Find [eq17].

nav_button Solution

First note that, by additivity:[eq18]

Therefore, in order to compute [eq17], we need to compute [eq20] and [eq21].

[eq20] is found using additivity on F:[eq23]so that:[eq24]

[eq25] is found using the fact that one minus the probability of an event is equal to the probability of its complement and the fact that [eq26]:[eq27]

As a consequence:[eq28]

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