Bag A contains 2 unbiased and 3 biased coins whereas Bag B contains 3 unbiased and 2 biased coins. A bag is selected at random and 2 coins are taken out simultaneously. The probability, that both coins are unbiased is:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(\displaystyle\frac{1}{5}\)
Each bag is chosen with probability \(\frac{1}{2}\). Two coins out of 5 can be chosen in \(^{5}C_{2} = 10\) ways.
\( \displaystyle P(\text{both unbiased} \mid \text{Bag A}) = \frac{^{2}C_{2}}{^{5}C_{2}} = \frac{1}{10} \)
\( \displaystyle P(\text{both unbiased} \mid \text{Bag B}) = \frac{^{3}C_{2}}{^{5}C_{2}} = \frac{3}{10} \)
By the theorem of total probability:
\( \displaystyle P = \frac{1}{2} \cdot \frac{1}{10} + \frac{1}{2} \cdot \frac{3}{10} = \frac{4}{20} = \frac{1}{5} \)
Why the other options are incorrect
Option (2) \(\displaystyle\frac{1}{10}\) — Incorrect. This is the probability for Bag A alone.
Option (3) \(\displaystyle\frac{2}{5}\) — Incorrect. This is \(\frac{1}{10} + \frac{3}{10}\), the sum of the two bag-wise probabilities without the factor \(\frac{1}{2}\) for choosing the bag.
Option (4) \(\displaystyle\frac{3}{10}\) — Incorrect. This is the probability for Bag B alone.