# 12. The Binomial Probability Distribution

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A **binomial** experiment is one that possesses the following properties:

The experiment consists of

*n*repeated trials;Each trial results in an outcome that may be classified as a

**success**or a**failure**(hence the name,**binomial**);The probability of a success, denoted by

*p*, remains constant from trial to trial and repeated trials are independent.

The number of successes *X* in *n* trials of a binomial experiment is called a **binomial random variable**.

The probability distribution of the random variable *X* is called a **binomial distribution**, and is given by the formula:

`P(X)=C_x^n p^x q^(n-x)`

where

n= the number of trials

x= 0, 1, 2, ...n

p= the probability of success in a single trial

q= the probability of failure in a single trial(i.e.

q= 1 −p)`C_x^n`

is a combination

*P*(*X*) gives the probability of successes in *n *binomial trials.

## Mean and Variance of Binomial Distribution

If *p* is the probability of success and *q* is the probability of failure in a binomial trial, then the expected number of successes in *n* trials (i.e. the mean value of the binomial distribution) is

E(X) =μ=np

The **variance** of the binomial distribution is

V(X) =σ^{2}=npq

Note: In a binomial distribution, only **2** parameters, namely *n* and *p*, are needed to determine the probability.

### Example 1

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A die is tossed `3` times. What is the probability of

(a) No fives turning up?

(b) `1` five?

(c) `3` fives?

### Example 2

Hospital records show that of patients suffering from a certain disease, `75%` die of it. What is the probability that of `6` randomly selected patients, `4` will recover?

### Example 3

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In the old days, there was a probability of `0.8` of success in any attempt to make a telephone call. (This often depended on the importance of the person making the call, or the operator's curiosity!)

Calculate the probability of having `7` successes in `10` attempts.

### Example 4

A (blindfolded) marksman finds that on the average he hits the target `4` times out of `5`. If he fires `4` shots, what is the probability of

(a) more than `2` hits?

(b) at least `3` misses?

### Example 5

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The ratio of boys to girls at birth in Singapore is quite high at `1.09:1`.

What proportion of Singapore families with exactly 6 children will have at least `3` boys? (Ignore the probability of multiple births.)

[**Interesting and disturbing trivia: **In most countries the ratio of boys to girls is about `1.04:1`, but in China it is `1.15:1`.]

### Example 6

A manufacturer of metal pistons finds that on the average, `12%` of his pistons are rejected because they are either oversize or undersize. What is the probability that a batch of `10` pistons will contain

(a) no more than `2` rejects? (b) at least `2` rejects?

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