Question-61
A bag contains \(5\) white 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)
\(66.7\)
Since the balls are drawn one at a time with replacement, we can assume that the draws are independent. Since the bag remains the same, we are also assured of identical distributions. Let \(X_{i}\) denote the random variable corresponding to the \(i^{\text{th}}\) draw that is \(1\) if a black ball is drawn and \(0\) otherwise. Then, \(X_{i} \sim \text{Bernoulli}( p)\), where \(p=\cfrac{10}{15} =\cfrac{2}{3}\). \(S_{n}\) can then be written as \(S_{n} =X_{1} +\cdots +X_{n}\). Therefore, \(S_{n} \sim \text{Binomial}( n,p)\). The expectation of \(S_{n}\) is \(E[ S_{n}] =np\). The required answer is \(E[ S_{100}] =\cfrac{200}{3} \approx 66.7\).