A bag contains 8 red and 7 black balls. Two balls are drawn at random. The probability that both the balls are of the same colour is:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \( \displaystyle\frac{7}{15} \)
There are 15 balls, so two balls can be drawn in \( ^{15}C_{2}=105 \) ways. Both balls are of the same colour if both are red or both are black:
\( \displaystyle ^{8}C_{2}=28,\qquad ^{7}C_{2}=21 \)
\( \displaystyle P(\text{same colour})=\frac{28+21}{105} \)
\( \displaystyle =\frac{49}{105}=\frac{7}{15} \)
Shortcut: Drawing one ball after the other, the two cases add directly:
\( \displaystyle \frac{8}{15}\cdot\frac{7}{14}+\frac{7}{15}\cdot\frac{6}{14}=\frac{56+42}{210} \)
\( \displaystyle =\frac{7}{15} \)