Question

Joe likes to go to an old arcade and play 8 games of Skeeball every day. On average, he wins a prize on 64% of the games he plays. Today, however, he won only 3 of the 8 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 64 % chance of winning each game he plays, like before. If he plays 8 games a day, on what percentage of days will he win 3 games or fewer?

Assume that each game is independent of the others.

Fill in the blank: Joe would win 3 games or fewer on __________ % of the days that he plays.

Help please with steps on how you solved it, I will rate and comment if answered.

Answer 1

Let X is a random variable shows the number of games he win. Here X has binomial distribution with parameters n=8 and p=0.64. The probability that Joe would win 3 games or fewer is

P(X < 3) = > C ) (0.64)"(1-0.64)"-エー0.1180

Answer: 11.80%