## Ask Professor Puzzler

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Today's question is a probability question from Sue. Her question is about how to calculate the probability of winning a game at least a certain number of times.

Let's say the probability of Sue winning Fraction Concentration against the computer is 3/4. If she plays 8 times, how do you calculate the probability that she'll win at least 6 times?

If she wins *at least* six times, that means she could win 6, 7, or 8 times.

So we're going to have to work out the probability that she wins exactly 6 times, the probability that she'll win exactly 7 times, and the probability that she'll win exactly 8 times. Once we've found those three probabilities, we add them together to get the total probability.

Note that in the equations below, 3/4 is the probability of a win, and 1/4 is the probability of a loss (since 1 - 3/4 = 1/4).

Six wins: (3/4)^6 x (1/4)^2

Seven wins: (3/4)^7 x (1/4)

Eight wins: (3/4)^8

Now, depending on who your teacher is, and whether or not they like to torture you with fractions, you'll either have an ugly fraction, or a decimal answer. I came up with approximately 0.1446.

Now that you know the process, you can fill in any numbers you like. Make the probiblity of winning 0.05. Make the number of games 20. Whatever you like.

Helpful tip: Suppose you played 20 games, and wanted the probability of winning at least 5. Using this method, you'd have to find the probability of winning 5, 6, 7, 8,...18, 19, or 20 games. Not pretty.

So instead, Find the probability of LOSING no more than 4. That's a much easier probability, and it comes out to the same thing!

Happy computing!

Professor Puzzler