You have 1,000 coins, and 995 of them are fair (equal probability of heads or tails). Five of them are weighted and have a 95% probability of landing on heads. You randomly choose one of the 1,000 coins. Find the probability that it is a weighted coin, under the following scenarios. (a) You flip it 10 times and it lands on heads 9 times (b) You flip it 20 times and it lands on heads 18 times

(a) 0.001575

(b) 0.009434

Step-by-step explanation:

(a) Let the event be P (W).

P (W) = P (weighted) * P (x = no of flips)

P (x) is given by a Bernoulli distribution and is given the probability of a head or tail landing during n flips

P (x) = nCr (p) ^r (q), q = 1-p

P = 95% = 0.95 q = 0.05. Hence for n=10 flips, r=9

P (W) = (5/1000) * 10C9 (0.95) ^9 (0.05) ¹

P (W) = 0.005 * 0.3151 = 0.001575

(b) P (W) = (5/1000) * 20C18 (0.95) ^18 (0.05) ^2

P (W) = 0.009434