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Chapter 7: Problem 26

Let \(E\) and \(F\) be two events of an experiment with sample space \(S\). Suppose \(P(E)=.6, P(F)=.4\), and \(P(E \cap F)=\) .2. Compute: a. \(P(E \cup F)\) b. \(P\left(E^{c}\right)\) c. \(P\left(F^{c}\right)\) d. \(P\left(E^{c} \cap F\right)\)

Short Answer

Expert verified
a. \(P(E \cup F) = 0.8\) b. \(P\left(E^{c}\right) = 0.4\) c. \(P\left(F^{c}\right) = 0.6\) d. \(P\left(E^{c} \cap F\right) = 0.2\)

Step by step solution

01

Recall the formula for probability of union

The probability of the union of two events E and F is given by: \(P(E \cup F) = P(E) + P(F) - P(E \cap F)\)
02

Plug in the given values and calculate

We are given \(P(E) = 0.6, P(F) = 0.4,\) and \(P(E \cap F) = 0.2.\) Plug these values into the formula: \(P(E \cup F) = 0.6 + 0.4 - 0.2 = 0.8\) b. Compute \(P\left(E^{c}\right)\)
03

Recall the formula for the probability of a complement

The probability of the complement of an event E is given by: \(P\left(E^{c}\right) = 1 - P(E)\)
04

Plug in the given value and calculate

We are given \(P(E) = 0.6.\) Plug this value into the formula: \(P\left(E^{c}\right) = 1 - 0.6 = 0.4\) c. Compute \(P\left(F^{c}\right)\)
05

Recall the formula for the probability of a complement

The probability of the complement of an event F is given by: \(P\left(F^{c}\right) = 1 - P(F)\)
06

Plug in the given value and calculate

We are given \(P(F) = 0.4.\) Plug this value into the formula: \(P\left(F^{c}\right) = 1 - 0.4 = 0.6\) d. Compute \(P\left(E^{c} \cap F\right)\)
07

Recognize the relationship between the given events

We can express \(P\left(E^{c} \cap F\right)\) as \(P(F) - P\left(E \cap F\right),\) because the intersection of \(E^c\) and F corresponds to the probability of F minus the common probability between E and F.
08

Plug in the given values and calculate

We are given \(P(F) = 0.4\) and \(P(E \cap F) = 0.2.\) Plug these values into the relationship: \(P\left(E^{c} \cap F\right) = 0.4 - 0.2 = 0.2\) To summarize, the answer to each part is a. \(P(E \cup F) = 0.8\), b. \(P\left(E^{c}\right) = 0.4\), c. \(P\left(F^{c}\right) = 0.6\), and d. \(P\left(E^{c} \cap F\right) = 0.2\).

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