4. Binomial Probability Formula

The probability of exactly 𝑘 successes is:

𝑃(𝑋 = 𝑘) = (𝑛 choose 𝑘)𝑝^𝑘(1−𝑝)^(𝑛−𝑘), where 0 ≤ 𝑘 ≤ 𝑛

• (𝑛 choose 𝑘) = (𝑛!)/(𝑘!(𝑛−𝑘)!)
• 𝑝^𝑘 = probability of those 𝑘 successes
• (1−𝑝)^(𝑛−𝑘) = probability of the remaining failures

The binomial probability formula gives us a way to calculate the probability of achieving exactly 𝑘 successes in 𝑛 independent trials, where each trial can result in a success with probability 𝑝 or a failure with probability (1−𝑝). The formula consists of three parts:

1. The binomial coefficient, represented by (𝑛 choose 𝑘), which calculates the number of different ways to choose which 𝑘 trials are successful out of 𝑛 total trials. This helps consider all possible combinations of successes.
2. The term 𝑝^𝑘 represents the probability of those successful outcomes occurring, raised to the power of the number of successes.
3. The term (1−𝑝)^(𝑛−𝑘) gives us the probability of the remaining trials resulting in failures. Together, the formula calculates the overall probability of having exactly 𝑘 successes among the trials.

Imagine you are flipping a coin 3 times, and you want to know the probability of getting exactly 2 heads (successes). Here, 𝑛 is 3 (the number of flips), 𝑘 is 2 (the desired number of heads), and 𝑝 is 0.5 (the probability of getting heads in one flip). You can use the formula to calculate:

• Find how many ways you can get 2 heads in 3 flips (which is 3).
• Then calculate the probability of getting those specific 2 heads and 1 tail. This exact process mirrors how you can model other situations with two outcomes using the binomial probability formula.

• (𝑛 choose 𝑘) = (𝑛!)/(𝑘!(𝑛−𝑘)!)

The binomial coefficient (𝑛 choose 𝑘) is a crucial part of the binomial probability formula and indicates how many different ways you can choose 𝑘 successes from 𝑛 trials. To calculate it, we use factorials:

• Factorial (𝑛!) represents the product of all positive integers up to 𝑛. For example, 5! = 5 x 4 x 3 x 2 x 1 = 120.
• So for our coefficient, we divide the factorial of 𝑛 by the product of the factorials of 𝑘 and (𝑛−𝑘). This division effectively eliminates duplicate sequences since the order of successes does not matter.

Think of it like choosing toppings for a pizza. If you can choose 2 toppings out of 3 options (let's say pepperoni, mushrooms, and olives), the different combinations you can create is like (3 choose 2). Using the formula, you'd find (3!)/(2!1!) = 3 ways to choose from those options: just pepperoni and mushrooms, just pepperoni and olives, or just mushrooms and olives. This illustrates that (𝑛 choose 𝑘) helps us understand how outcomes can be organized without caring about their order.

• 𝑝^𝑘 = probability of those 𝑘 successes
• (1−𝑝)^(𝑛−𝑘) = probability of the remaining failures

In the binomial probability formula, the terms 𝑝^𝑘 and (1−𝑝)^(𝑛−𝑘) represent the probabilities of successes and failures, respectively. Here’s how to break it down:

• 𝑝^𝑘: This part calculates the likelihood of having exactly 𝑘 successes (where each success has a probability 𝑝). Each success raises this probability term to the power of the number of successes, accurately reflecting how likely these successes are to occur together.
• (1−𝑝)^(𝑛−𝑘): Conversely, this term calculates the likelihood of the remaining trials all being failures. It raises the failure probability (which is simply 1 minus the success probability) to the power equal to the number of failures (𝑛−𝑘). Together, these two factors allow the formula to cover all aspects of a scenario involving binary outcomes.

Consider a basketball player who makes a free throw 70% of the time (𝑝 = 0.7). If they shoot 5 times and you want to know the probability of making exactly 3 shots, you can quantify this with:

• Probability of making 3 shots (𝑝^3 = 0.7^3), and
• Probability of missing 2 shots (1−𝑝 = 0.3, thus (0.3)^2). This mirrors how success and failure interrelate when calculating probabilities across trials.

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