Andy has 4 red socks and 8 black socks in his drawer. He takes 2 socks at random from his drawer.
- Find the probability that the socks taken are of different colours.
The random variable X is the number of red socks taken.
- Draw up the probability distribution table for X.
- Find E(X).
Solution
(i) To find the probability that the socks taken are of different colours, we can use the formula:
\(P(\text{different colours}) = P(RB) + P(BR)\)
Where:
- \(P(RB) = \frac{4}{12} \times \frac{8}{11}\)
- \(P(BR) = \frac{8}{12} \times \frac{4}{11}\)
Thus,
\(P(\text{different colours}) = \frac{4}{12} \times \frac{8}{11} + \frac{8}{12} \times \frac{4}{11} = \frac{16}{33}\)
(ii) The probability distribution table for \(X\), the number of red socks taken, is:
| Number of red socks | 0 | 1 | 2 |
|---|
| Prob | \(\frac{14}{33}\) | \(\frac{16}{33}\) | \(\frac{3}{33}\) |
(iii) To find \(E(X)\), the expected value of \(X\), we use:
\(E(X) = 0 \times \frac{14}{33} + 1 \times \frac{16}{33} + 2 \times \frac{3}{33}\)
\(E(X) = \frac{16}{33} + \frac{6}{33} = \frac{22}{33} = \frac{2}{3}\)
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