The probability that you will win a game is $p = 0.85$.
- If you play the game $1294$ times, what is the most likely number of wins?
- Let $X$ represent the number of games (out of $1294$) that you win. Find the standard deviation for the probability distribution of $X$.
(a) Notice that this resembles the Binomial Distribution with $n=1294,$ $p=0.85,$ and $q=0.15.$ To find the most likely number of wins, simply compute the mode, which is either $\lfloor (n+1)p\rfloor$ or $\lceil (n+1)p-1\rceil.$ In this case, there are two maximum values since the solution $k$ to $\frac{(n-k)p}{(k+1)(1-p)}=1$ is not an integer.
(b) The standard deviation is determined by the formula $\sqrt{np(1-p)}.$
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