A fair coin is tossed and a fair die is thrown. The probability of getting a head on the coin or a six on the die is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(\displaystyle\frac{7}{12}\)
Let H be the event “a head on the coin” and S the event “a six on the die”. The coin and the die do not affect each other, so H and S are independent:
\( \displaystyle P(H\cap S) = \frac{1}{2}\times\frac{1}{6} = \frac{1}{12} \)
By the addition theorem:
\( P(H\cup S) = P(H)+P(S)-P(H\cap S) \)
\( \displaystyle P(H\cup S) = \frac{1}{2}+\frac{1}{6}-\frac{1}{12} = \frac{6+2-1}{12} = \frac{7}{12} \)
Why the other options are incorrect
Option (2) \(\displaystyle\frac{2}{3}\) — Incorrect. This is \(\frac{1}{2}+\frac{1}{6}\); the case “head and six” is counted twice.
Option (3) \(\displaystyle\frac{1}{12}\) — Incorrect. This is the chance of a head and a six together.
Option (4) \(\displaystyle\frac{5}{12}\) — Incorrect. This is \(\frac{1}{2}\times\frac{5}{6}\), the chance of neither a head nor a six.