Tennis Anyone? To win a game in tennis, a player must win...

Tennis Anyone? To win a game in tennis, a player must win four points. If both players have won three points, the play continues until a player is ahead by two points to win the game. The model $$ P(x)=\frac{x^{4}\left(-8 x^{3}+28 x^{2}-34 x+15\right)}{2 x^{2}-2 x+1} $$ represents the probability $P$ of a player winning a game in which the player is serving the game and $x$ is the probability of winning a point on serve. The player serving is the first to put the ball in play. Source: Chris Gray, "Game, set and stats," Significance, February $2015 .$ (a) What is the probability that a player who is serving will win the game if the probability of the player winning a point on serve is $0.64 ?$ (b) Find and interpret $P(0.62)$. (c) Solve $P(x)=0.9$ (d) Graph $P=P(x)$ for $0 \leq x \leq 1$. Describe what happens to $P$ as $x$ approaches 1 .

Tennis Anyone? To win a game in tennis, a player must win four points. If both players have won three points, the play continues until a player is ahead by two points to win the game. The model
$$
P(x)=\frac{x^{4}\left(-8 x^{3}+28 x^{2}-34 x+15\right)}{2 x^{2}-2 x+1}
$$
represents the probability $P$ of a player winning a game in which the player is serving the game and $x$ is the probability of winning a point on serve. The player serving is the first to put the ball in play. Source: Chris Gray, "Game, set and stats," Significance, February $2015 .$
(a) What is the probability that a player who is serving will win the game if the probability of the player winning a point on serve is $0.64 ?$
(b) Find and interpret $P(0.62)$.
(c) Solve $P(x)=0.9$
(d) Graph $P=P(x)$ for $0 \leq x \leq 1$. Describe what happens to $P$ as $x$ approaches 1 .

Key Concepts

Solving Nonlinear Equations

Solving nonlinear equations, particularly those derived from probabilistic models in sports, involves finding the value of a variable that satisfies the equation. In these problems, one may need to determine the specific probability of winning a point on serve that leads to a desired overall game-winning probability, thus necessitating techniques for handling polynomial equations.

Graphical Analysis and Limit Behavior

Graphical analysis involves plotting a function to observe its behavior across its domain. In the context of probability models, this includes examining how the function behaves as the underlying parameter approaches its extreme values, such as 0 or 1. Analyzing the limit behavior as the probability of winning a point on serve approaches 1 provides insights into how game-winning probability converges, often reflecting the intuition that a near-certain point win results in a near-certain game win.

Probability in Sequential Games

This concept involves calculating the likelihood of winning a game, where each point is an independent event with its own probability of success. In sequential games like tennis, where a player must win a set number of points and sometimes overcome tie situations (such as deuce), the overall probability of winning the game is determined by considering the composite outcomes of multiple point-by-point events.

Polynomial Modeling in Sports

Polynomial models are used to represent complex probabilistic processes in sports by encapsulating the impact of multiple sequential events into a single algebraic expression. In this context, the probability of winning a tennis game is expressed as a ratio of polynomials in the variable representing serving advantage, reflecting the cumulative probability of winning successive points while accounting for extended play scenarios.

Transcript

00:01 With a probability formula for tennis, really the important point to know here is that x is the probability of winning a point on serve.

00:16 So p is the probability of a player winning a game in which the player is serving the game, and x is the probability of winning the point on the serve.

00:25 So i'm just going to write that x equals probability of winning point on serve.

00:37 That is important.

00:40 So they've given us this equation, and for part a, they've asked us, what is the probably of the player winning a point on serve? what is it probably that a player who is serving will win the game if the probability of a player winning a point on the serve is 0 .64? important part about that is pretty much they're telling us x is 0 .64, because they've told us the probability of winning the point on the serve is 0 .64.

01:08 So all we are doing is substituting 0 .64 into the equation.

01:16 Everywhere there's next.

01:18 I could rewrite that, but, well, okay, i guess i will.

01:22 0 .64 power 4 minus 8 times 0 .64 cubed plus 28 times 0 .64 squared minus 34 times 0 .64 squared, minus 34 times 0 .64, plus 15.

01:40 All divided by 2 times 0 .64 squared minus 2 times 0 .64 plus 1.

01:51 So throwing that into our calculator, that's going to take a little bit here, but just make sure that you've got your brackets in the right place, minus 34 times 0 .64, plus 15, all divided by 2 .0 .2 .2.

02:15 0 .64 squared minus 2 times 0 .64 plus 1.

02:21 Awesome.

02:23 Okay, so throwing in our value here of 0 .64.

02:27 We have a probability of 0 .81.

02:35 So we have about an 81 % chance that the person serving is winning the game.

02:48 So what is it probably the third? okay.

02:51 So now it says find and interpret probability of 0 .62.

02:58 So i'm doing the exact same thing.

03:00 I'm just plugging in 0 .62 instead of 0 .64 and then interpreting my results.

03:07 So exact same method pretty much just swap out for 0 .62 instead of 0 .64.

03:13 And you end up with 0 .78.

03:18 So what that means is that.

03:21 There's a 0 .62 probability of winning the point on the serve, which then gives you a 0 .878 probability of the player winning a game in which the player is serving the game.

03:40 So if their probability is good, they're going to win a point on the serve, then the probability is good that they're going to win the game.

03:47 It's kind of how that works.

03:50 So then it says for c and d we want to graph it.

03:54 It has a graphing icon next to it.

03:57 So we're going to solve it.

04:00 And really that's the easiest way to solve part c where we have to solve for probability to find x is by graphing.

04:08 So we are going to throw this lovely monstrosity of an equation into our graphing program...

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