Mathematically Modelling Basketball Shots

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Mathematically Modelling Basketball Shots

Situation

The manager of a professional basketball team is having a tough decision in choosing which of his two top scorers this season are better at free-throw shots. The final decision will go towards picking the team for Saturday’s Cup Final match.

        On a training session one week before the match the coach decides to “go all out” and bring some mathematical genii in to model a situation where Lee Grimes and Dominic Aspbury, the goalscorers, will shoot at the basketball net.

        The mathematical genii are students from Cambridge and are benefiting from this opportunity in that they will be able to show evidence of coursework for their final exam. Their coursework will be using their abilities to collect data and “test the appropriateness of a probability model” on a real situation whilst the coach’s aim will be to pick the better of the two players for the “big game.”

        

If the random variables X and Y count the number of independent trials before the event, having a probability p, occurs then X and Y have geometric distributions:

                        

P ( X = r ) = q r – 1 p        where  r = 1,2,3,……

X~G ( p )           and         Y~G ( p )

I will define X         as being the number of shots required before Lee shoots a basket. Therefore, Y is defined as the number of shots required before Dom shoots a basket. I will be attempting to see if X and Y have geometric distributions by taking samples of X and Y.

The populations are the infinite range of shots capable from the two throwers taken in a discrete time period under varied conditions at the same level of skill. This is impossible to create so the coursework will have to involve sampling, therefore not producing results representative of the whole population. For this coursework I can not take random samples because it will not be possible to recreate due to the infinite choices of shot which could occur e.g. fatigue levels, mood type, improvement of skill level throughout the sampling etc. all could differ.

        I will record a sample of X by asking Lee to shoot a number of baskets and hence work out the relative frequency of success – ‘p’. This result will allow me to model X as X~G ( p ) . Next I will record a sample for Y by asking Dom to shoot a number of baskets so that another value for the relative frequency of success – ‘p’ can be calculated. I can use the result to model Y as Y~G ( p ) .

        The conditions I will have to use are going to be as similar as possible to gain independent and identical shots. This will involve:

  • Five practice shots beforehand so that the feel of shooting is apparent – a warm up before starting.
  • The shots being taken from the same free-throw position which is fifteen feet away from the base of the net and perpendicular to the back line.
  • The same type of shot being used – using one hand to steady the ball and one to project the ball through the air. Same arms used each time.
  • The weather conditions being similar. In the sports hall there should be no significant alteration of the environment.
  • Each shot being taken one after the other to gain results, which will be under the most similar conditions.
  • When the shot is taken; a score implies one basket, a no score implies try again until you succeed.
  • Continue until the sample of eighty is reached and record all results

If the data is successful I may be able to produce a reliable geometric model of the population from the sample enabling me to predict population parameters with greater confidence. Using the parameters I should be able to compare the populations by considering sample parameters. I have chosen a geometric model because it is an infinite distribution requiring discrete random variables and is able to accommodate the infinite range of shots that may be required to score a basket. The sum of all the probabilities will equal one (a probability density function).

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If X and Y have a geometric distribution, the distribution should look like this:

The sample size shall be 80 as a large sample size makes the geometric distribution as accurate as possible for testing purposes. It also allows me to use the chi squared test on the model to check if there is any evidence to suggest that one thrower is better than the other at various critical levels.

        Assumptions that I am making to allow the model to work are that the trials are:

  • Identical: The factors are exactly the same. This ...

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