Show all work Considering the random experiment of tossing a

Show all work

Considering the random experiment of tossing a balanced coin three times and observing the result (a head H and a tail T) for each toss. Let A be the event exactly two heads are tossed, B be the event that the first toss is a head, and C be the event that the second toss is a head. Determine the following probabilities:

a) the probability that the first toss is a head.

b) the conditional probability that the first toss is a head , given that exactly two heads are tossed.

c) the conditional probability that the first toss is a head given that the second toss is a head.

Solution

a)

Events are independent. Thus,

P(first is head) = 1/2 [answer]

regardless of the net tosses.

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b)

Those with exactly 2 heads are

HHT
HTH
THH

Among these 3, 2 have a heads as first toss. Thus,

P(first is head|2 heads) = 2/3 [answer]

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c)

Again, events are independent. The second toss does not influence the first toss.

Hence,

P(first is head|seconf is head) = P(first is head) = 1/2 [answer]