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.

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.

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