Probability of winning a game

Tennis Outcomes Related to Probabilities of Winning Individual Points

Suppose in every point of a match you had a 50% probability of winning every point. If you played thousands of such matches you would expect to win as many games, sets, and matches as you lose. Furthermore, you expect that much more of the time the scores will be close, like 6-4 rather than 6-0. What are the scores like and how often do you win for other probabilities of winning each point? The following graphs show the results for relevant probabilities. For example, if your probability of winning each point were 70%, your probability of winning a game is 91%, you virtually never lose a set, and 60% of the time you win a set at 6-0.

These results were generated by the Python computer program shown below that used a random number generator to “play” points according to the different probabilities and which then kept score, using a 7 point tie-breaker at 6-6. For each probability, 10,000 matches were played. An interesting result is that even with a very modest improvement in one’s probability of winning a point, one’s probability of winning the match is very dramatically improved. For example, if one’s probability of winning a point improves from 50% to 55%, the probability of winning a set improves from 50% to 84%.

As we get to smaller units, the statistical averages become less helpful in making strategic decisions. Nonetheless, at the game level, if one’s probability of winning a point is 0.5, 0.6, 0.7, and 0.8, one’s probability of winning a game is 0.5, 0.74, 0.91, and 0.98. It is also of interest to know in a partially completed game, what the probability is of winning the game. The following table shows the probability that A wins the game starting from the indicated points for player A and B. For example, from the score 30-40, A has a 25% chance of winning the game. Note that from a score of 40-Love, between equal players, A should win the game 95% of the time. In actual play, it seems like A actually wins the game less than 90% of the time. This discrepancy indicates that either or both A and B do not maintain the same level of intensity and concentration independent of the score.

Probability of winning a game Tennis Outcomes Related to Probabilities of Winning Individual Points Suppose in every point of a match you had a 50% probability of winning every point. If you## Applying Probability to Games of Chance

In a typical finite math course, you will have many opportunities to study games that involve chance—where you can’t control the play results, but you can be informed about the possibilities.

The following three examples use Bunco, a lottery, and a poker hand to explore these opportunities and their probabilities.

### Bunco

Consider the game of Bunco, where you roll three dice. What is the probability that you roll the three dice and get all fours?

The probability of rolling a four is 1/6. You have three dice involved, so to find the probability of all three rolls resulting in a four, use the multiplication property and get

This is about 0.5%. The chance is slight. If you’d be happy with having the three dice be the same, not caring about which number, then you add the six possibilities: either three ones, three twos, three threes, and so on:

This is about 3%—better than just fours but still not very likely.

### Lottery

Your state has a weekly lottery, and you really want to win the big prize. All you have to do is come up with the same six numbers that the machine on television does. You choose from the numbers 1 through 60. Your choice is based on your birthday: December 24, 1960, at 6:52 a.m. Your numbers will be 12, 24, 19, 60, 6, and 52. What are the chances you’ll win?

First, you count up how many different ways the six numbers can be chosen. The order doesn’t matter—they put them in order after they’re drawn. To count the number of ways to choose 6 numbers from 60, you use the formula for combinations:

This comes out to be 50,063,860 different ways to choose the six numbers. Your chance of winning:

Not looking good …

### Poker

If you’re a poker player, then you know that a flush is a good hand to have. If you don’t know anything about poker, don’t worry. This is fairly easy to describe.

A poker player is dealt five cards. If all five are the same suit, then she has a flush. What is the chance of being dealt a flush? A deck of cards has four different suits (spades, hearts, diamonds, and clubs), and each suit has 13 different cards. So, a flush could be five spades, five hearts, five diamonds, or five clubs. This sounds much easier than six out of 60 numbers!

To find the probability of being dealt a flush, you count how many ways you can be dealt a flush and then divide that by how many different five-card hands are possible. The order doesn’t matter, so you use combinations.

To be dealt a flush in a suit, you need to have five of those 13 cards. So, using combinations,

There are four different suits to choose from, so multiply that result by 4 and get 5,148 different ways you can get a flush.

Now, determine how many different five-card hands can be dealt. That’s a combination of 52 cards taken five at a time.

Divide the number of ways you can get a flush by the total number of hands and you have

or about 0.2% chance of being dealt a flush. Of course, with draw poker, you can try to improve your original hand if it isn’t what you want at first.

Applying Probability to Games of Chance In a typical finite math course, you will have many opportunities to study games that involve chance—where you can’t control the play results, but you can ]]>