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# Finding Probability with Given Restrictions

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12. Let E, F, and G be three events. Find expressions for the events so that of E, F, and G:

(a) only E occurs;
(b) both E and G but not F occur;
(c) at least one of the events occurs;
(d) at least two of the events occur;
(e) all three occur;
(f) none of the events occurs.

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#### Solution Preview

Let the events E, F and G be independent of each other

Let the probability of occurrence of E be denoted by P(E)
Then the probability of E not occurring = Q(E) = {1-P(E)}

Let the probability of occurrence of F be denoted by P(F)
Then the probability of F not occurring = Q(F) = {1-P(F)}

Let the probability of occurrence of G be denoted by P(G)
Then the probability of G not occurring = Q(G) = {1-P(G)}

a) Only E occurs;

This means that E occurs and F and G do not occur
Probability of E occurring and F and G not occurring =
P(E) x Q (F) x Q (G)
= P(E) x {1-P (F)} x {1-P (G) }
Answer: P(E) x {1-P (F)} x {1-P (G) }

b) Both E and G but ...

#### Solution Summary

The solution gives expressions for the probability of the events.

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