A bag contains 5 white balls and 10 black balls. In a random experiment, \(n\) balls are drawn from the bag one at a time with replacement. Let \(S_{n}\) denote the total number of black balls drawn in the experiment.
The expectation of \(S_{100}\) denoted by \(E[S_{100}] =\) ________
(Round off to one decimal place)
Answer
66.6 to 66.7
Explanation
Drawing with replacement leaves the bag unchanged, so the draws are independent and each is black with probability \(\frac{10}{15} = \frac{2}{3}\).
Write \(S_{100} = Y_{1} + \cdots + Y_{100}\), where \(Y_{k} = 1\) if the \(k\)-th draw is black and 0 otherwise. Each \(E[Y_{k}] = \frac{2}{3}\), so by linearity of expectation
\[ E[S_{100}] = 100 \times \frac{2}{3} = \frac{200}{3} \approx 66.67 \]
This is the mean \(np\) of a Binomial\((100, \frac{2}{3})\) variable. Rounded to one decimal place it is 66.7; using the white-ball probability \(\frac{1}{3}\) by mistake would give 33.3.
Answer: 66.7