1 problem found
In Fridge football, each team scores two points for a goal and one point for a foul committed by the opposing team. In each game, for each team, the probability that the team scores \(n\) goals is \(\left(3-\left|2-n\right|\right)/9\) for \(0\leqslant n\leqslant4\) and zero otherwise, while the number of fouls committed against it will with equal probability be one of the numbers from \(0\) to \(9\) inclusive. The numbers of goals and fouls of each team are mutually independent. What is the probability that in some game a particular team gains more than half its points from fouls? In response to criticisms that the game is boring and violent, the ruling body increases the number of penalty points awarded for a foul, in the hope that this will cause large numbers of fouls to be less probable. During the season following the rule change, 150 games are played and on 12 occasions (out of 300) a team committed 9 fouls. Is this good evidence of a change in the probability distribution of the number of fouls? Justify your answer.
Solution: \begin{array}{c|c|c|c} k & \P(k \text{ goals}) & \P(\geq 2k+1 \text{ fouls}) & \P(k \text{ goals and } \geq 2k+1 \text{ fouls}) \\ \hline 0 & \frac{3-|2|}{9} = \frac19 & \frac{9}{10} & \frac{9}{90}\\ 1 & \frac{3-|2-1|}{9} = \frac29 & \frac{7}{10} & \frac{14}{90} \\ 2 & \frac{3-|2-2|}{9} = \frac39 & \frac{5}{10} & \frac{15}{90} \\ 3 & \frac{3-|2-3|}{9} = \frac29 & \frac{3}{10} & \frac{6}{90} \\ 4 & \frac{3-|2-4|}{9} = \frac19 & \frac{1}{10} & \frac{1}{90} \\ \hline &&& \frac{9+14+15+6+1}{90} = \frac12 \end{array} The probability a team scores more than half its points from fouls is \(\frac12\). Letting \(X\) be the number of times a team committed \(9\) fouls, then \(X \sim B(300, p)\). Consider two hypotheses: \(H_0: p = \frac1{10}\) \(H_1: p < \frac1{10}\) Under \(H_0\), we are interested in \(\P(X \leq 9)\). Since \(300 \frac{1}{10} > 5\) it is appropriate to use a normal approximation, \(N(30, 27)\). Therefore, \begin{align*} && \P(X \leq 9) &\approx \P(3\sqrt{3}Z + 30 \leq 9.5) \\ &&&= \P( Z \leq \frac{9.5-30}{3\sqrt{3}}) \\ &&&= \P(Z \leq \frac{-20.5}{3\sqrt{3}}) \\ &&&< \P(Z \leq -\frac{7}{2}) \end{align*} Which is very small. Therefore there is good evidence to believe there has been a change in the number of fouls.