(Solved): 2. A random experiment is repeated \( n \) times. In each experiment, the probability of success i ...

2. A random experiment is repeated \( n \) times. In each experiment, the probability of success is \( p \). The outcomes of the experiment are observed and recorded. Let \( X \) be the number of the observed successes. a) Write down the probability \( p_{k}=\operatorname{Pr}(X=k) \). Describe the general trend of \( p_{k} \). b) Determine the (integer) value of \( k \) such that \( p_{k} \) is a maximum for a fixed \( n \). c) Determine the condition that there exist two possible (integer) values of \( k \) yielding the same maximum..