A box contains two coins: a fair coin and a coin with a head on both sides. One coin is chosen at random from the box and tossed once. The probability that it shows a head is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(\displaystyle\frac{3}{4}\)
Let \(E_1\) be the event that the fair coin is chosen and \(E_2\) the event that the two-headed coin is chosen, and let H be the event of a head.
\( \displaystyle P(E_1) = P(E_2) = \frac{1}{2} \)
\( \displaystyle P(H|E_1) = \frac{1}{2},\quad P(H|E_2) = 1 \)
By the theorem of total probability:
\( \displaystyle P(H) = \frac{1}{2}\times\frac{1}{2}+\frac{1}{2}\times 1 \)
\( \displaystyle = \frac{1}{4}+\frac{1}{2} = \frac{3}{4} \)
Why the other options are incorrect
Option (1) \(\displaystyle\frac{1}{2}\) — Incorrect. This is the chance of a head with the fair coin only.
Option (2) \(\displaystyle\frac{2}{3}\) — Incorrect. This is \(P(E_2|H)\), the probability that the two-headed coin was chosen, given a head.
Option (4) 1 — Incorrect. This is the chance of a head with the two-headed coin only.