Geometric Series
A Sports Application

In this module we look at a simple model of tennis. Suppose that you are playing tennis against a familiar opponent. The two of you have always played very consistently. You always make 85% of your shots and your opponent always makes 80% of her shots. For this simple model we will ignore the differences between serving and ordinary shots. The drawing below shows what happens on the first few shots when you serve first.

If you miss the first shot then you lose. If you make the first shot then there are two possibilities -- either your opponent misses her first shot and then she loses OR your opponent makes her first shot. If she makes her first shot then there are two new possibilities -- either you miss your second shot and lose or you make your second shot. If you make your second shot then there are two new possibilities. ...

The drawing below shows the probability of each of the various possibilities.

Notice that the probability of you missing your first shot is 0.15 -- resulting in an immediate loss.
The probability of you making your first shot is 0.85. If this happens then there are two possibilities.
- Your opponent misses her first shot. This happens with probability 0.20, so the net probability of you making your first shot followed by your opponent missing her first shot is (0.85) * (0.20) = 0.17. In this case you win.
- Your opponent makes her first shot. This happens with probability 0.80, so the net probability of you making your first shot followed by your opponent making her first shot is (0.85) * (0.80) = 0.68. In this case the point continues.

Looking at the first two shots we see there are three possibilities.

You lose on your first shot -- this happens with probability 0.15.
You win on your opponent's first shot -- this happens with probability 0.17
The point continues -- this happens with probability 0.68.

This gives us our first estimates for the outcome of this point

You win -- probability 0.17.
You lose -- probability 0.15.
Nobody wins -- probability 0.68.

We use the following notation

W(n) -- the probability that you win considering n shots by each player. Notice W(1) = 0.17.
L(n) -- the probability that you lose considering n shots by each player. Notice L(1) = 0.15.
T(n) -- the probability of neither side winning after n shots by each player. Notice T(1) = 0.68.

Notice that


            Lim   W(n) = Probability that you win the point.
          n --> oo

            Lim   L(n) = Probability that you lose the point.
          n --> oo

            Lim   T(n) = Probability that the point goes on forever.
          n --> oo

Find W(2), L(2), and T(2).
Find W(3), L(3), and T(3).
Find formulas for W(n), L(n), and T(n). Notice that two of these are geometric series.
Find the probability that you win the point and the probability that your opponent wins the point..
Answer the questions above for a point in which your opponent makes the first shot.
If you flip a coin to determine who chooses whether to make the first shot and you win the coin toss should you choose to make the first shot?
You have been debating whether to play a more aggressive game. If you play a more aggressive game then your probability of making each shot goes down to 0.70 but your opponent's probability of returning each shot goes down to 0.60. Discuss the pros and cons of this new strategy. Notice that your new strategy does not affect the probability of your opponent making her first shot if she makes the first shot -- it only affects the probability of her making each return.

Geometric Series A Sports Application

Geometric Series
A Sports Application