Ximpledu

# Binomial Distribution

See how to solve a binomial distribution:
finding P(X), E(X), V(X), σ(X).
3 examples and their solutions.

## Binomial Experiment

### Definition

B(n, p)
A binomial experiment is
repeating an independent event.
n: Number of repeating the event
p: Probability of the event

### Formula

P(X = k) = nCkpkqn - k
P(X = k): Probability of the 'want' event happening k times.
nCk: n trials, the number of ways to pick 'want' k times
q = 1 - p
(Probability of 'not want')
Binomial Theorem
Combination (Math)

### Example

A coin is tossed 7 times.
Solution

### Example

A fair die is tossed 3 times.
P(at most one '4') = ?
Solution

## Binomial Distribution

### Definition

B(n, p)

If n is big enough,
the probability histogram shows
the shape of a binomial distribution.
Shape: Normal distribution

### Formula

B(n, p)

E(X) = np
V(X) = npq
σ(X) = npq
E(X): Expected value
V(X): Variance
σ(X): Standard deviation
Variance, Standard Deviation

### Example

A fair die is tossed 180 times.

1. Find the expected value of the number of getting a '2'.
2. Find the variance of the number of getting a '2'.
3. Find the standard deviation of the number of getting a '2'.
Solution