12. The Binomial Probability Distribution

A binomial experiment is one that possesses the following properties:

  1. The experiment consists of n repeated trials;

  2. Each trial results in an outcome that may be classified as a success or a failure (hence the name, binomial);

  3. 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) = σ2 = npq

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

Example 1

die
Image source

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

telephone
Image source

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 imortance 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

baby
Image source

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?

Didn't find what you are looking for on this page? Try search:

Online Algebra Solver

This algebra solver can solve a wide range of math problems. (Please be patient while it loads.)

Ready for a break?

 

Play a math game.

(Well, not really a math game, but each game was made using math...)

The IntMath Newsletter

Sign up for the free IntMath Newsletter. Get math study tips, information, news and updates each fortnight. Join thousands of satisfied students, teachers and parents!

Given name: * required

Family name:

email: * required

See the Interactive Mathematics spam guarantee.

Share IntMath!

Short URL for this Page

Save typing! You can use this URL to reach this page:

intmath.com/binomprob

Math Lessons on DVD

 

Easy to understand math lessons on DVD. See samples before you commit.

More info: Math videos

Loading...
Loading...