Problem 9 You have a biased coin for which PH p You toss th

Problem 9 You have a biased coin for which P(H) = p. You toss the coin 20 times. What is the probability that a. you observe 8 heads and 12 tails; b. you observe more than 8 heads and more than 8 tails?

Solution

a)

Using the binomial distribution formula,

P(8 heads) = 20C8 p^8 (1 - p)^(20-8)

= 125970 p^8 (1 - p)^12 [ANSWER]

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We can have more than 8 heads and more than 8 tails with these results:

9H, 11T
10H, 10T
11H, 9T

Thus, we use the binomial probability formula to get the probability of each case, then add them up.

Thus,

P(more than 8 heads and more than 8 tails) = 20C9 p^9 (1 - p)^(20 - 9) + 20C10 p^10 (1 - p)^(20 - 10) + 20C11 p^11 (1 - p)^(20 - 11)

= 167960 p^9 (1 - p)^11 + 184756 p^10 (1 - p)^10 + 167960 p^11 (1 - p)^9 [ANSWER]