practice_problems/problems/unit-01/statistical-independence/18

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practice_problems/problems/unit-01/statistical-independence/18

https://mids-w203.github.io/practice_problems/problems/unit-01/statistical-independence/18.html

[jps245]

a. P(two-headed | H) = p(two-headed-heads and H) / p(H)
p(H) = p(two-headed-tails and H) + p(fair coin and H) + p(two-headed-heads and H) = (0)(1/3) + (1/2)(1/3) + (1)*(1/3) = 1/2
p(two-headed-heads and H) = 1/3
P(two-headed | H) = 1/3 / 1/2 = 2/3

b. after the first head, we have the updated probabilties for coins;
p(two-headed-heads) = 2/3 (see a.)
p(two-headed-tails) = 0
p(fair coin) = 1 - 2/3 - 0 = 1/3
p(2H) = p(two-headed-heads) * p(H|two-headed-heads) + p(fair coin) * p(H|fair coin) = 2/31 + 1/31/2 = 5/6 (where 2H = 2 heads in a row)

c. P(two-headed | 2H) = p(two-headed-heads and 2H) / p(2H)
P(2H) = 4/6 + 1/6 = 5/6
P(2H and two-headed-heads) = 4/6
P(two-headed | 2H) = 4/6 /5/6 = 4/5

Please let me know if you disagree. Thanks