A box contains 3 coins. One coin has 2 heads and the other two are fair. A coin is chosen at random from the box and flipped. If the coin turns up heads, what is the probability that it is the two-headed coin? Is the answer 1/3? Was the answer intuitive?

Accepted Solution

Answer: Our required probability is [tex]\dfrac{1}{2}[/tex]Step-by-step explanation:Since we have given that Number of coins = 3Number of coin has 2 heads = 1Number of fair coins = 2Probability of getting one of the coin among 3 = [tex]\dfrac{1}{3}[/tex]So, Probability of getting head from fair coin = [tex]\dfrac{1}{2}[/tex]Probability of getting head from baised coin = 1Using "Bayes theorem" we will find the probability that it is the two headed coin is given by[tex]\dfrac{\dfrac{1}{3}\times 1}{\dfrac{1}{3}\times \dfrac{1}{2}+\dfrac{1}{3}\times \dfrac{1}{2}+\dfrac{1}{3}\times 1}\\\\=\dfrac{\dfrac{1}{3}}{\dfrac{1}{6}+\dfrac{1}{6}+\dfrac{1}{3}}\\\\=\dfrac{\dfrac{1}{3}}{\dfrac{2}{3}}\\\\=\dfrac{1}{2}[/tex]Hence, our required probability is [tex]\dfrac{1}{2}[/tex]No, the answer is not [tex]\dfrac{1}{3}[/tex]