GATE Overflow Test Series | Probability | Test 1 | Question: 26

Question

Two coins are in a box. The coins look alike, but one coin is fair (with equal probabilities for Head and Tail), while the other coin is biased, with $1/3$ probability of Heads. One of the coins is randomly chosen from the hat, without knowing which of the two it is. Now this chosen coin is tossed twice, showing Heads both times. Given this information, what is the probability that the chosen coin is the fair coin?

• A. $\dfrac{4}{5}$
• B. $0.692$
• C. $\dfrac{1}{4}$
• D. $\dfrac{1}{2}$

Answer 1

By using Bayes theorem for conditional probability we get

$P\left(\frac{\text{fair}}{HH}\right) =\frac{ P\left(\frac{HH}{\text{fair}}\right) \times P\left(\text{fair}\right)}{P\left(\frac{HH}{\text{biased}}\right)\times P\left(\text{biased}\right)+ P\left(\frac{HH}{\text{fair}}\right) \times P\left(\text{fair}\right)}$

$\qquad=\frac{\frac{1}{4}\times\frac{1}{2}}{\frac{1}{9}\times\frac{1}{2}+\frac{1}{4} \times\frac{1}{2}} = 0.6923$

Answer 2

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