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)`


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

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

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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 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

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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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