5 Kirk and Les regularly play each other at darts.
- The probability that Kirk wins any game is 0.3 , and the outcome of each game is independent of the outcome of every other game.
Find the probability that, in a match of 15 games, Kirk wins:
- exactly 5 games;
- fewer than half of the games;
- more than 2 but fewer than 7 games.
- Kirk attends darts coaching sessions for three months. He then claims that he has a probability of 0.4 of winning any game, and that the outcome of each game is independent of the outcome of every other game.
- Assuming this claim to be true, calculate the mean and standard deviation for the number of games won by Kirk in a match of 15 games.
- To assess Kirk's claim, Les keeps a record of the number of games won by Kirk in a series of 10 matches, each of 15 games, with the following results:
$$\begin{array} { l l l l l l l l l l }
8 & 5 & 6 & 3 & 9 & 12 & 4 & 2 & 6 & 5
\end{array}$$
Calculate the mean and standard deviation of these values.
- Hence comment on the validity of Kirk's claim.