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STATLECT > Fundamentals of probability > Independent events

Independent events - Exercise set 1

This exercise set contains some solved exercises on independent events. The theory needed to solve these exercises is introduced in the lecture entitled Independent events.

Exercise 1.1

Suppose that we toss a die. Six numbers (from 1 to $6)$ can appear face up, but we do not yet know which one of them will appear. The sample space is:[eq1]Each of the six numbers is a sample point and is assigned probability $ rac{1}{6}$. Define the events E and F as follows:[eq2]Prove that E and F are independent.

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The probability of E is:[eq3]The probability of F is:[eq4]The probability of $Ecap F$ is:[eq5]E and F are independent because:[eq6]

Exercise 1.2

A firm undertakes two projects, A and $B$. The probabilities of having a successful outcome are $ rac{3}{4}$ for project A and $ rac{1}{2}$ for project $B$. The probability that both projects will have a successful outcome is $ rac{7}{16}$. Are the two outcomes independent?

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Denote by E the event 'project A is successful', by F the event 'project $B$ is successful' and by $G$ the event 'both projects are successful'. The event $G$ can be expressed as:[eq7]If E and F are independent, it must be that:[eq8]Therefore, the two outcomes are not independent.

Exercise 1.3

A firm undertakes two projects, A and $B$. The probabilities of having a successful outcome are $ rac{2}{3}$ for project A and $ rac{4}{5}$ for project $B$. What is the probability that neither of the two projects will have a successful outcome if their outcomes are independent?

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Denote by E the event 'project A is successful', by F the event 'project $B$ is successful' and by $G$ the event 'neither of the two projects is successful'. The event $G$ can be expressed as:[eq9]where $E^{c}$ and $F^{c}$ are the complements of E and F. Using De Morgan's law ([eq10]) and the formula for the probability of a complement, we obtain:[eq11]Using the formula for the probability of a union, we obtain:[eq12]Finally, since E and F are independent:[eq13]

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