Question

Joe likes to go to an old arcade and play 12 games of Skeeball every day. On average, he wins a prize on 55% of the games he plays. Today, however, he won only 4 of the 12 games he played.

This made Joe think that there must be something wrong with how he is throwing the ball today. However, a friend tells him, "You might be throwing the same way you usually do. People will sometimes have a group of bad games just because of random variation, not because of psychology, or because they are doing anything differently, or any other reason. For someone with your statistics, such random bad days wouldn't even be very rare."

Let's see what Joe's friend means. Suppose that Joe's skill level really hasn't changed, so he still has a 55% chance of winning each game he plays, like before. If he plays 12 games a day, on what percentage of days will he win 4 games or fewer? Assume that each game is independent of the others.

Step 1

In this context, there are fixed number of trials. That is, *n* = 12. There are two possible outcomes (Won the game and Lose the game), outcomes are independent, and probability of success is fixed for each trial. Thus, the random variable *x* follows binomial distribution with *n* = 12 and *p* = 0.55.

Step 2

Let *x* denotes number of games won by Joe and *n* denotes total number of games played. To get the percentage of days that Joe would win 4 games or fewer is obtained using the Binomial formulae:

Step 3

The percentage of days that Joe would win 4 games or few...

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