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Chapter Contents

Counting and Probability - Introduction 1. Factorial Notation 2. Basic Principles of Counting 3. Permutations 4. Combinations 5. Introduction to Probability Theory 6. Probability of an Event Singapore TOTO Probability and Poker 7. Conditional Probability 8. Independent and Dependent Events 9. Mutually Exclusive Events 10. Bayes’ Theorem 11. Probability Distributions - Concepts 12. Binomial Probability Distributions 13. Poisson Probability Distribution 14. Normal Probability Distribution The z-Table Counting and Probability Problem Solver

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12. The Binomial Probability Distribution

A binomial experiment is one that possesses the following properties:

On this page...

Mean and variance of a binomial distribution

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) = Cn xpx qn−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)

Cn x 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

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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?

Answer

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?

Answer

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.

Answer

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?

Answer

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.]

Answer

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?

Answer

 11. Probability Distributions - Concepts

13. Poisson Probability Distribution 

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