# Concept

• Success/failure
• Detection of a disease is a success!

## Example

The probability of choosing statistics is 0.16. If 20 students are randomly selected, what is the probability that

• at least 5 would choose the subject?
• exactly 2 would choose the subject?
• none will choose?

## Function

$$P(x) = ^nC_x \space p^x(1-p)^{n-x}$$

$$1-p=q$$

• $$p \to$$ probability of success
• $$q \to$$ probability of failure
• $$n \to$$ number of Bernoulli trials
• $$x \to$$ no. of success

## Bernoulli Trial

• Introduced by James Bernoulli
• Independent trials with dichotomous outcomes (success/failure)
• Coin toss, disease detection

## Bernoulli Distribution

$$p^xq^{1-x}$$

If n = 1 in binomial distribution

# Assumptions of Binomial Distribution

• Trials are independent
• p, q remain constant for each trial
• Outcomes are dichotomous
• X counts number of success

## Properties of Binomial Distribution

• Mean: $$np$$
• Variance: $$npq$$
• Skewness, $$\beta_1 = \frac{(q-p)^2}{npq}$$
• Kurtosis, $$\beta_2 = 3 + \frac{1-6pq}{npq}$$

## Derivation of pmf

$$P(SF SF \cdots) = P(S) \times P(F) \cdots$$

• $$(p \times p \cdots \times p)(\times q \times q \cdots \times q)$$
• $$p^xq^{n-x}$$
• $$^nC_x \space p^xq^{n-x}$$ (any x combination from n)
• 2 items from a, b, c $$\to ab, ac, bc \to \space ^3C_2$$

## Derivation of Mean

$$\displaystyle E(X) = \sum_{x=0}^n x\cdot p(x)$$ (think: why zero?)

• $$0 \cdot p(0) + \cdots$$
• $$np\{q^{n-1}+n(n-1)pq^{n-2}+\cdots+p^{n-1}\}$$
• $$np(q+p)^{n-1} [binomial \ theorem]$$
• $$np$$ [p+q=1]

## Uses of Binomial Distribution

• Quality control
• Test of fit

See here

## Binomial to Noraml

If $$n \to \infty$$ and $$np\ge 5, nq \ge 5$$

## Problem 01

The mean and sd of a binomial distribution are 40 and 6. Find

1. $$n,p,q$$
2. $$P(x)$$
3. $$P(x=0), P(x\ge2), P(x\lt3)$$
• $$np = 40$$
• $$\sqrt{npq} = 6$$
• $$np(1-p)=36 \to 40(1-p)=36 \to 1-p=.9$$

## Problem 02

An unbiased coin is tossed 10 times. Find the probability of

## Problem 03

The probability that it rains on a particular day is 0.2. If 10 days are observed, what is the probability that it would rain on

1. all 10 days
2. no days
3. at least 2 days
4. on exactly four days

## Problem 04

In El Clasico, the probability that Barcelona wins is 0.39 and that Real Madrid wins is 0.42. In the next 5 matches, what is the probability that

1. Barcelona wins all the matches?
2. Real wins all the matches
3. Barcelona wins at least 2 matches
4. 2 matches are drawn
5. more than 3 matches are drawn
• $$p_1=$$Barca wins, $$p_2 =$$ Real wins, $$p_3 =$$Match drawn