Example of Binomial Distribution

Example Data

Suppose we flip a fair coin 10 times. Each flip has:

• Two possible outcomes: Heads (H) or Tails (T)
• Probability of getting Heads (p) = 0.5
• Probability of getting Tails (q) = 1 - p = 0.5

The number of heads in 10 flips follows a binomial distribution.

1. Probability Calculation

Q: What is the probability of getting exactly 6 heads in 10 flips?

Using the Binomial Probability Formula:

P(X = k) = C(n, k) * p^k * (1 - p)^(n - k)
    

Where:

• n = 10 (total trials)
• k = 6 (successes - heads)
• p = 0.5 (probability of heads)

First, compute the binomial coefficient:

C(10, 6) = 10! / (6!(10-6)!) = 210
    

Now, calculate the probability:

P(X = 6) = 210 * (0.5)^6 * (0.5)^4
         = 210 * (0.015625) * (0.0625)
         = 0.2051
    

So, the probability of getting exactly 6 heads in 10 flips is 20.51%.

2. Visual Representation

The binomial distribution for 10 flips of a fair coin would look like:

📊 Probability Mass Function (PMF) Representation

• The most probable outcomes are around 5 heads (mean of distribution).
• The distribution is symmetric since p = 0.5.
• Probabilities decrease as we move towards extreme values (0 or 10 heads).

3. Real-World Applications in AI

• A/B Testing: Used to measure success rates in experiments.
• Predicting customer behavior: Probabilities of purchase decisions.
• Machine learning classification: Bernoulli trials for modeling binary outcomes.