Binomial Probability Basics for Sports Bettors

You do not need a math background to work out how likely a shooter is to hit three of five free throws. Binomial probability is the tool, and this beginner lesson builds it from the ground up. First you read a probability as a number from 0 to 1, then separate a one-time event from a repeated one. The key question is whether attempts are independent, meaning one result does not change the next. From there you count makes and misses step by step and check your math on a simple shooter example. Prices, including a 62 cent prediction market contract, turn out to be probabilities in disguise, which makes the skill useful well beyond props.

Learning objectives

By the end, you will be able to read a probability as a number between 0 and 1, tell a one-time event from a repeated one, and use the binomial idea to work out how likely a set of makes and misses is. No math background needed. Every new word gets a plain definition.

What a probability is

A probability is a number from 0 to 1 that says how likely something is. A 0 means it cannot happen. A 1 means it must happen. A 0.5 is a coin flip. You will also see the same number written as a percent, so 0.4 is 40%.

Betting prices are probabilities in disguise. Say a prediction market contract costs 62 cents and pays $1 if the event happens. That price points to about 62%. Read it as what traders are saying, not as a fact about the world. You can practice reading prices this way on the prediction markets page.

One event versus many events

A single event has one outcome you care about, like the home team wins or the player scores. When one of several outcomes must happen, their probabilities add up to 1. If a team has a 0.58 chance to win and a tie is impossible, the other side has 0.42.

Lots of bets are really small events stacked together. A shooter takes five free throws. A hitter gets four at-bats. To handle those, ask one question first: are the attempts independent? Independent means one result does not change the next one. A coin flip is independent. A shooter who gets tired late in a game may not be.

The binomial idea

Binomial is a fancy word for a simple setup. You have a fixed number of tries. Each try has two results, such as made or missed. The chance is the same every try, and the tries are independent. The model then tells you the chance of getting exactly some number of successes.

There are three steps. First, find the chance of one specific order, like make, make, miss. Second, count how many different orders give the same total. Third, multiply the two. The count of orders is written C(n,k) and read "n choose k". For 3 tries with 2 makes, there are 3 orders: make-make-miss, make-miss-make, and miss-make-make.

Counts without a fixed number of tries

Some things are counts with no set number of attempts, like goals in a soccer match or strikeouts in a game. A Poisson model uses one number, the average count, to estimate how likely 0, 1, 2 or more events are. You do not need to calculate it by hand yet. Just know the shape. The average sits in the middle, and real results scatter around it, often wider than the model expects.

Worked example

Dana tracks a made-up shooter named Luis, who makes 70% of his free throws. He takes 3 shots. Dana asks: what is the chance he makes exactly 2?

1. The chance of a make is 0.7. The chance of a miss is 0.3.
2. One order, make-make-miss: 0.7 x 0.7 = 0.49, then 0.49 x 0.3 = 0.147.
3. Three orders give exactly 2 makes: 3 x 0.147 = 0.441.

So the chance is 0.441, or 44.1%.

Now check the whole picture. Zero makes: 0.3 x 0.3 = 0.09, then 0.09 x 0.3 = 0.027. One make: 0.7 x 0.09 = 0.063, then 3 x 0.063 = 0.189. Three makes: 0.7 x 0.7 = 0.49, then 0.49 x 0.7 = 0.343. Add all four: 0.027 + 0.189 + 0.441 + 0.343 = 1. They add to 1, so the math holds together.

One caution. The model treats Luis as the same 70% shooter on every shot. If he tires or the defense changes, the real answer moves.

Practice

Exercise 1. A shooter makes 50% of his shots and takes 2. What is the chance he makes both?

Answer: 0.5 x 0.5 = 0.25, or 25%.

Exercise 2. Flip a fair coin twice. What is the chance of exactly one head? There are 2 orders, head-tail and tail-head.

Answer: each order is 0.5 x 0.5 = 0.25, and 2 x 0.25 = 0.5, or 50%.

Recap

1. A probability is a number from 0 to 1, and the chances of all possible outcomes of one event add up to 1.
2. The binomial setup needs a fixed number of tries, two results, the same chance each time, and independent tries.
3. To find the chance of an exact count, multiply the chance of one order by the number of orders, then check that all counts add to 1.

Frequently asked questions

How do you calculate binomial probability in sports?

Count the tries, set the chance of success on each, then multiply the chance of one order by the number of orders that give your total. For 3 shots at 70% with exactly 2 makes, that is 3 x 0.147 = 0.441.

What does independent mean in probability?

Independent means one result does not change the chance of the next. A coin flip is independent. Shots by a tired player may not be, since fatigue can affect later attempts. The binomial model only works well when tries are close to independent.

Can a probability be more than 1?

No. A probability runs from 0, which cannot happen, to 1, which must happen. If your numbers for every possible outcome of one event add up to more or less than 1, something in the math or the inputs is off.

Related lessons

• Machine Learning for Sports Forecasting
• NBA Three-Point Props: Modeling Attempts and Makes

Sources and further reading

• About Sharp App.

Try it in the Prediction markets screen. Examples in this lesson are fictional, built to show the method. Work the numbers by hand first, then repeat the same steps on a real board. You can also read how Prediction Markets works on the Sharp App site.

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