You have 100 coins, and 99 of them are fair (equal probability of heads or tails
ID: 3073234 • Letter: Y
Question
You have 100 coins, and 99 of them are fair (equal probability of heads or tails). One of them is weighted and has a 90% probability of landing on heads. You randomly choose one of the 100 coins. Find the probability that it is a weighted coin, under the following scenarios: (Hint: if your calculator can’t compute 100!, R can, just type factorial(100))
(a) You flip it 10 times and lands on heads 10 times (b) You flip it 10 times and it lands on heads 9 times
(c) You flip it 20 times and it lands on heads 18 times (d) You flip it 100 times and it lands on heads 77 times
Explanation / Answer
(a) Here we have 100 coins, 99 of them are fair and one of them is weightd and 90% probability of landing on heads.
We flip it 10 times and lands on heads 10 times
Pr(Weighted coin l flip it 10 times and lands on heads 10 times )
Probability of any coin landing heads 10 times out of 10 times
Pr(landing 10 time head on 10 times) = Pr(Normal coin) * Pr(Normal Coin landing 10 heads on 10 times) + Pr(Weighted Coin) * Pr(weighted Coin landing 10 heads on 10 times) = 99/100 * (0.5)10 + 1/100 * (0.9)10
Pr(Weighted Coin) = 1/100 * (0.9)10 / [99/100 * (0.5)10 + 1/100 * (0.9)10]
= 0.7829
(b) Pr(Flip 10 times and lands 9 timess) = Pr(Normal coin) * Pr(Normal Coin landing 9 heads on 10 times) + Pr(Weighted Coin) * Pr(weighted Coin landing 9 heads on 10 times)
= 99/100 * 10C9 (0.5)10 + 1/100 * 10C9 (0.9)9 (0.1)
Pr(Weighted coin) = [1/100 * 10C9 (0.9)9 (0.1) ]/ [99/100 * 10C9 (0.5)10 + 1/100 * 10C9 (0.9)9 (0.1) ]
= 0.2861
(c) Here flip it 20 times and lands on heads 18 times
= Pr(Normal coin) * Pr(Normal Coin landing 18 heads on 20 times) + Pr(Weighted Coin) * Pr(weighted Coin landing 18 heads on 20 times)
= 99/100 * 20C18 (0.5)20 + 1/100 * 20C18 (0.9)18 (0.1)2
Pr(Weighted coin) = [1/100 * 20C18 (0.9)18 (0.1)2 ]/ [99/100 * 20C18 (0.5)20 + 1/100 * 20C18 (0.9)18 (0.1)2 ]
= 0.9408
(d) Here flip it 100 times and it lands on 77 times
Pr(Head on 77 times out of 100) =
Pr(Normal coin) * Pr(Normal Coin landing 77 heads on 100 times) + Pr(Weighted Coin) * Pr(weighted Coin landing 77 heads on 100 times)
= 99/100 * 100C77 (0.5)100 + 1/100 * 100C77(0.9)77 (0.1)23
Pr(Weighted coin) = [1/100 * 100C77(0.9)77 (0.1)23 ]/ [99/100 * 100C77 (0.5)100 + 1/100 * 100C77(0.9)77 (0.1)23 ]
= 0.9746
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