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Chapter 1: Q7E (page 21)

Consider two events A and B with Pr(A) = 0.4 and Pr(B) = 0.7. Determine the maximum and minimum possible values of \(Pr\left( {A \cap B} \right)\) and the conditions under which each of these values is attained.

Short Answer

Expert verified

The maximum value is\(\Pr \left( {A \cap B} \right) = 0.4\)

The minimum value is\(\Pr \left( {A \cap B} \right) = 0.28\)

Step by step solution

01

Given information

The probability of an event A is

\(\Pr \left( A \right) = 0.4\)

The probability of an event B is:

\(\Pr \left( B \right) = 0.7\)

02

Computing the required probability

\(\Pr \left( {A \cap B} \right)\) is maximum when \(A \subset B\)

In such a case,

\(\begin{aligned}{}\Pr \left( {A \cap B} \right) &= \Pr \left( A \right)\\\Pr \left( {A \cap B} \right) &= 0.4\end{aligned}\)

When two events, A and B, are independent \(\Pr \left( {A \cap B} \right)\) is minimum.

The probability is:

\(\begin{aligned}{}\Pr \left( {A \cap B} \right) &= \Pr \left( A \right)\Pr \left( B \right)\\ &= 0.4 \times 0.7\\ &= 0.28\end{aligned}\)

Thus the required probability is:

\(\Pr \left( {A \cap B} \right) = 0.28\)

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