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# Probability

See the attached file.
1) Let A and B be two events.
a) If the events A and B are mutually exclusive, are A and B always independent? If the answer is no, can they ever be independent? Explain
b) If A is a subset of B, can A and B ever be independent events? Explain

2) Flip an unbiased coin five independent times. Compute the probability of
a) HHTHT
b) THHHT
c) HTHTH
d) Three heads occurring in the five trials.

3) An urn contains two red balls and four white balls. Sample successively five times at random and with replacement, so that the trials are independent. Compute the probability of each of the two sequences WWRWR and RWWWR.

#### Solution Preview

Please see the attachment for response.

Another method for question 5.12 d

If a coin is tossed five times, three heads can occur in 5C3 = 10 ways.
Thus the probability of three heads occurring in five trials = (5C3)*(1/2)^5 = 10*(1/2)^5 =5/16.

Solution 5.11

a) If the events A and B are mutually exclusive, then and hence .
If the events A and B are independent, then and hence
and
.
That is the probability of A occurring must not be affected by the fact that B has occurred, and vice versa.
Thus if the ...

#### Solution Summary

The solution contains a detailed explanation of the determination of probabilities in various situations.

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