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Chapter 7: Problem 99

Explain how the addition principle for mutually exclusive events follows from the general addition principle.

Short Answer

Expert verified
The addition principle for mutually exclusive events follows from the General Addition Principle by considering that mutually exclusive events have no overlap, meaning their intersection is the empty set (\(P(A \cap B) = 0\)). This condition is applied to the General Addition Principle formula, resulting in a simplified formula for mutually exclusive events: \(P(A \cup B) = P(A) + P(B)\).

Step by step solution

01

Definition of Mutually Exclusive Events

Mutually exclusive events are events in which the occurrence of one event excludes the occurrence of the other event(s). In other words, if event A occurs, event B cannot occur, and vice versa. Mathematically, this means that the intersection of the events is the empty set: \(P(A \cap B) = 0\).
02

Definition of General Addition Principle (GAP)

The General Addition Principle is a rule that states the probability of the occurrence of either event A or event B, or both, can be calculated as follows: \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\). It takes into account the intersection of events, which represents the overlap between events A and B.
03

Identify Mutual Exclusiveness in the GAP

If events A and B are mutually exclusive, there is no overlap between them, which means that their intersection is equal to zero (\(P(A \cap B) = 0\)). Therefore, we can apply this condition to the General Addition Principle formula.
04

Apply the Mutual Exclusiveness Condition to the GAP

Given the condition that \(P(A \cap B) = 0\), we can rewrite the General Addition Principle formula for mutually exclusive events as follows: \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\) where \(P(A \cap B) = 0\). So, the formula becomes: \(P(A \cup B) = P(A) + P(B) - 0\).
05

Simplify the Resulting Formula

After applying the condition, we can further simplify the formula to get the addition principle for mutually exclusive events: \(P(A \cup B) = P(A) + P(B)\)
06

Conclusion

The addition principle for mutually exclusive events follows from the General Addition Principle by taking into account the fact that there is no overlap between the two events (i.e., their intersection is the empty set). When the intersection is empty, the general formula simplifies to the addition principle for mutually exclusive events.

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Most popular questions from this chapter

Based on the following table, which shows the performance of a selection of 100 stocks after one year. (Take S to be the set of all stocks represented in the table.) $$ \begin{array}{|r|c|c|c|c|} \hline & \multicolumn{3}{|c|} {\text { Companies }} & \\ \cline { 2 - 4 } & \begin{array}{c} \text { Pharmaceutical } \\ \boldsymbol{P} \end{array} & \begin{array}{c} \text { Electronic } \\ \boldsymbol{E} \end{array} & \begin{array}{c} \text { Internet } \\ \boldsymbol{I} \end{array} & \text { Total } \\ \hline \begin{array}{r} \text { Increased } \\ \boldsymbol{V} \end{array} & 10 & 5 & 15 & 30 \\ \hline \begin{array}{r} \text { Unchanged }^{*} \\ \boldsymbol{N} \end{array} & 30 & 0 & 10 & 40 \\ \hline \begin{array}{r} \text { Decreased } \\ \boldsymbol{D} \end{array} & 10 & 5 & 15 & 30 \\ \hline \text { Total } & 50 & 10 & 40 & 100 \\ \hline \end{array} $$ If a stock stayed within \(20 \%\) of its original value, it is classified as "unchanged." Find all pairs of mutually exclusive events among the events \(P, E, I, V, N\), and \(D .\)

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