Free Throw Probability Calculator

A Free Throw Probability Calculator is a statistical tool that determines the likelihood of various outcomes when a basketball player takes a series of free throws. It goes beyond a player's simple free throw percentage to answer more complex questions. For example, it can calculate the probability of a player making an exact number of shots (e.g., "exactly 2 out of 3"), or the probability of making "at least" or "at most" a certain number of shots. By using the principles of binomial probability, this calculator is an essential tool for coaches, sports analysts, and fans who want to understand game situations with greater analytical depth, helping to evaluate end-of-game strategies and player consistency.

formula of Free Throw Probability Calculator

The calculation of free throw probability relies on a player's historical performance and the principles of binomial probability.

1. Single Free Throw Probability (The Core Input)

This is the fundamental data point you need. It is the shooter's historical probability of making a single free throw, expressed as a decimal.
Formula:
p = Free Throw Percentage / 100

• p: The probability of success (making a shot).
  The probability of missing a single free throw is complementary to this.
  Formula:
  q = 1 - p
• q: The probability of failure (missing a shot).

2. Probability of a Specific Sequence

This calculates the probability of a specific outcome happening in a precise order (e.g., Make, Miss, Make).
Formula:
P(Sequence) = p^k * q^(n-k)

• P(Sequence): The probability of the specific sequence occurring.
• k: The number of shots made in the sequence.
• n: The total number of shots taken.

3. Probability of Making an Exact Number of Shots (in Any Order)

This is the most common free throw probability calculation. It uses the binomial probability formula to find the chances of making exactly 'k' shots out of 'n' total attempts, regardless of the order.
Formula:
P(X=k) = C(n, k) * p^k * q^(n-k)

• P(X=k): The probability of making exactly 'k' shots.
• C(n, k): The number of combinations (also known as "n choose k"), which calculates how many different ways 'k' makes can be arranged in 'n' attempts.

3.1. Combination Formula (n Choose k)

This is a required sub-calculation for the main formula above.
Formula:
C(n, k) = n! / (k! * (n-k)!)

• n! ("n factorial"): The product of all positive integers up to n (e.g., 4! = 4 * 3 * 2 * 1 = 24).

4. Probability of Making "At Least" or "At Most" a Certain Number

To calculate these probabilities, you sum the probabilities of each individual outcome.

• Probability of Making AT LEAST 'k' Shots
  Formula: P(X>=k) = P(X=k) + P(X=k+1) + ... + P(X=n)
• Probability of Making AT MOST 'k' Shots
  Formula: P(X<=k) = P(X=0) + P(X=1) + ... + P(X=k)

Probability of Making Both Shots in a Two-Shot Foul

This table provides a quick reference for the probability of a player making both free throws based on their overall free throw percentage.

Player's Free Throw %	Probability of Making Both Shots (p²)
60%	36%
70%	49%
75%	56.25%
80%	64%
85%	72.25%
90%	81%
95%	90.25%

Example of Free Throw Probability Calculator

Let's calculate the probability of an 80% free throw shooter making exactly 2 out of 3 shots in a crucial end-of-game situation.

First, we define our variables.

• Probability of making a shot (p): 80% = 0.8
• Probability of missing a shot (q): 1 - 0.8 = 0.2
• Total number of shots (n): 3
• Target number of made shots (k): 2

Step 1: Calculate the number of combinations C(n, k).
C(3, 2) = 3! / (2! * (3-2)!)
C(3, 2) = (3 * 2 * 1) / ((2 * 1) * 1!) = 6 / 2 = 3
This means there are 3 possible sequences for making exactly 2 out of 3 shots: (Make-Make-Miss), (Make-Miss-Make), and (Miss-Make-Make).

Step 2: Use the binomial probability formula.
P(X=2) = C(3, 2) * p^k * q^(n-k)
P(X=2) = 3 * 0.64 * 0.2 = 0.384

To express this as a percentage, we multiply by 100.
The probability is 38.4%.

Therefore, an 80% free throw shooter has a 38.4% chance of making exactly two out of three free throws.

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