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

Answer 1

(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