(a)
Let Y be the number of heads on X number of flips.
Y | X ~ Binomial(n = X, p = 0.7)
Probability that the coin was flipped six times = Probability of no heads when the coin was flipped six times + Probability of exactly one head when the coin was flipped six times + Probability of exactly one head when the coin was flipped five times * Probability of head when the coin was flipped 6th time
= 0.03078
(b)
Given that coin was flipped six times, probability that two heads appeared in the six flips
= Probability of exactly one head when the coin was flipped five times * Probability of head when the coin was flipped 6th time / Probability that the coin was flipped six times
= 0.6447368
(c)
The support of X is 2, 3, 4, 5, 6 (We need at least 2 flips wo get two heads)
The probability distribution of X is,
P(X = 3) = Probability of exactly one head when the coin was flipped two times * Probability of head when the coin was flipped 3rd time
P(X = 4) = Probability of exactly one head when the coin was flipped three times * Probability of head when the coin was flipped 4th time
P(X = 5) = Probability of exactly one head when the coin was flipped four times * Probability of head when the coin was flipped 5th time
The probability dostribution of X is,
X | 2 | 3 | 4 | 5 | 6 |
P(X) | 0.49 | 0.294 | 0.1323 | 0.05292 | 0.03078 |
(d)
E(X) = 2 * 0.49 + 3 * 0.294 + 4 * 0.1323 + 5 * 0.05292 + 6 * 0.03078 = 2.84048
(e)
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