Expected Number of Attempts to Make 10 Free Throws in a Row: A Deep Dive

The task of determining the expected number of attempts a player needs to make 10 consecutive free throws in a row with an 80% shooting accuracy is a fascinating problem in probability theory. This article explores the application of geometric distributions and Markov chains to solve this problem and provides an in-depth look into the calculations required. Whether you're a sports enthusiast or a student of statistics, this guide will provide valuable insights.

Defining the Problem

We are given that the player has an 80% (or p 0.8) chance of making a free throw and a 20% (q 1 - p 0.2) chance of missing it. Our goal is to find the expected number of attempts needed to achieve 10 successful free throws in a row. To approach this, we'll use the concept of states and Markov chains to model the problem.

Modeling with States

We can model the problem using a series of states based on the number of consecutive successful free throws made:

State 0: No successful throws in a row (starting state) State 1: 1 successful throw in a row … State 9: 9 successful throws in a row State 10: 10 successful throws in a row (absorbing state, where the process stops)

Expected Attempts from Each State

Let E_n be the expected number of attempts needed to reach the State 10 from State n. The equations for the expected attempts can be defined as follows:

For n 10: E_{10} 0, as reaching this state means the player has already achieved the goal. For n 10:

If the player makes the next shot, they move to state n 1; if they miss, they go back to State 0. The equation for E_n can be written as:

E_n 1 p cdot E_{n 1} q cdot E_0

For the base case, for n 0:

E_0 1 p cdot E_1 q cdot E_0

Rearranging the equation for E_0 gives:

E_0 (1 - q) 1 p cdot E_1 E_0 frac{1 p cdot E_1}{1 - q}

This recursive relationship allows us to calculate E_0 by first finding E_1, E_2, …, E_9, and then substituting back these values.

Recursive Calculation

By continuing this process, we can derive a series of equations for each E_n from n 1 to 9. These equations can be solved iteratively or recursively to find E_0.

Expected Number of Attempts Calculation

Interestingly, there is also a direct formula for the expected number of trials needed to achieve k successes in a row, given by:

E frac{p^k}{1 - p^k} cdot (1 k)

Substituting p 0.8 and k 10, we get:

E approx frac{0.8^{10}}{1 - 0.8^{10}} cdot (1 10)

Calculating 0.8^{10}:

0.8^{10} approx 0.1074

And 1 - 0.8^{10}:

1 - 0.8^{10} approx 0.8926

Substituting these values into the formula:

E approx frac{0.1074}{0.8926} cdot 11 approx 1.079 cdot 11 approx 11.87

Finally, multiplying by the reciprocal of p 0.8 gives the expected number of attempts:

E approx frac{11.87}{0.8} approx 14.84

Conclusion

Based on these calculations, the expected number of attempts before a player with an 80% shooting accuracy makes 10 successful free throws in a row is approximately 14.84 attempts. These methods, using geometric distributions and Markov chains, provide a powerful way to model and solve sequential success problems in probability.