A winning streak in sports describes a situation where a player wins several games in a row. Human intuition is remarkably poor when it comes to judging the chance of such winning streaks. We tend to underestimate how often a long run of successes will happen purely by chance. The famous Sports Illustrated jinx says that sportspeople who feature on the cover of the prestigious Sports Illustrated magazine will soon have a crash in performance. It isn’t really a jinx: it’s just that front-page success stories are often nothing more than chance winning streaks, which inevitably come to an end.

Let X be the random variable that describes the maximum length of winning streak you’ll have during the course of 10 games, if your chance of winning each game is just 50%. The probability function of X is below, with one probability missing.

a. Find the missing probability p.

b. Write down the CDF, FX(x), in tabular format.

c. Give sketches of the probability function and the CDF.
Make sure your axes are labelled correctly, and ensure you have chosen a suitable type of graphic for each plot.

d. Find the probability of each of the following events. Each of your answers should have three parts: the appropriate probability statement in terms of X on the left, its expression in terms of fX(x) or FX(x) in the middle, and the calculated value on the right. For example, if the event specified corresponded to X = 9, your answer would be P(X = 9) = fX(9) = 0.002.
i. Your best winning streak in ten games is two consecutive wins.

es 557 ii. At some point during the ten games, you have a streak of at least five consecutive wins.
iii. You don’t win any of the ten games.
iv. You win all ten of the games.
v. You never win as many as four games in a row.