Binomial Distribution Calculator

What is Binomial Distribution?

The binomial distribution is a discrete probability distribution that models the number of successes in a fixed number of independent Bernoulli trials. Each trial has the same probability of success.

Formula and Its Components

The probability mass function of the binomial distribution is given by:

\[P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\]

Where:

• \(n\) is the number of trials
• \(k\) is the number of successes
• \(p\) is the probability of success on each trial
• \(\binom{n}{k}\) is the binomial coefficient, calculated as \(\frac{n!}{k!(n-k)!}\)

Calculation Steps

1. Determine the values of n, k, and p from the given problem.
2. Calculate the binomial coefficient \(\binom{n}{k}\).
3. Compute \(p^k\) and \((1-p)^{n-k}\).
4. Multiply these three terms together to get the final probability.

Example Calculation

Let's calculate the probability of getting exactly 3 heads in 5 coin tosses (assuming a fair coin).

• n = 5 (total number of tosses)
• k = 3 (number of heads we want)
• p = 0.5 (probability of getting heads on a single toss)

Step 1: Calculate \(\binom{5}{3} = \frac{5!}{3!(5-3)!} = 10\)

Step 2: Calculate \(0.5^3 = 0.125\)

Step 3: Calculate \(0.5^{5-3} = 0.5^2 = 0.25\)

Step 4: Multiply: \(10 \times 0.125 \times 0.25 = 0.3125\)

Therefore, the probability of getting exactly 3 heads in 5 tosses of a fair coin is 0.3125 or 31.25%.

Visual Representation

This bar graph represents the binomial distribution for 5 coin tosses. Each bar represents the probability of getting a specific number of heads, from 0 to 5.