KS5. S1. Binomial and Geometric Distributions

Binomial Distribution

A binomial distribution occurs where there are multiple trials, where each outcome can be success or failure.

We use combinations, ⁿC_k, to count the number of ways to get k successes in n trials.

If the probability of success at an individual trial is p, then the probability of failure is q = 1-p

Formula (to memorise)

Necessary conditions to use binomial distribution:
- Fixed number of trials, n;
- Only two possible outcomes (success / failure);
- Fixed probability of success, p;
- Trials are independent of each other

(N.B. In exam questions these conditions must be related to the context of the questions).

Worked Example

A regular pentagonal spinner is shown. Find the probability that 10 spins produce exactly three As.

Worked Example 2

The random variable X is distributed B(12, 1/6). Find:
- P(X=2)
- P(X=9)
- P(X≤1)

Worked Example 3

The probability that a randomly chosen member of a class is left-handed is 0.15. A random sample of 20 members of the class is taken.
1. Suggest a suitable model for the random variable X, the number of members in the sample who are left-handed. Justify your choice.
2. Use your model to calculate the probability that:
  1. Exactly 7 of the members in the sample are left-handed;
  2. Fewer than 2 of the members in the sample are left-handed.

Exercise

Answers

Expectation and Variance of Binomial Distribution

It is possible to write out the distribution and calculate it as in the previous exercise. Or:

Based on symmetry, we have that E(X) = np and Var(X) = np(1-p)

Worked Example 1

Given that X ~ B(12, 0.3), find the mean, the variance and the standard deviation of X .

Worked Example 2

The random variable X ~B(n, p). Given that E(X ) = 12 and Var(X ) = 7.5, find:

a) the value of n and of p;
b) P(X=11).

Exercise

Answers

Geometric Distribution

Imagine that we are trying to roll a “1” with a fair 6-sided die.

How likely is it that we would succeed on the first time? What about the second time? etc.

P(1 on first roll) = p

P(1 on second roll) = (1-p)p

P(1 on 3rd roll) = (1-p)²p

The following table shows the probabilities of success in this kind of “geometric” distribution:

r	1	2	3	4	…	n	…
P(X=r)	p	p(1-p)	p(1-p)²	p(1-p)³	…	p(1-p)⁴	…

Note that the values of P(X=r) in the above table are in geometric sequence with first term p and common ration (1-p), so their sum to infinity is $\frac{p}{1-(1-p)} = \frac{p}{p} = 1$ . This makes the geometric distribution suitable as a probability distribution, because the sum of the probabilities is 1.

Necessary conditions to use geometric distribution:
- Repeated trials are independent of each other;
- Repeated trials can be infinite in number;
- Each trial has exactly two possible outcomes (success / failure);
- Probability of success in each trial, p, is constant.

Formula

Worked Example 1

Repeated independent trials are carried out in which the probability of success in each trial is 0.66. Correct to 3 significant figures, find the probability that the first success occurs:
- on the third trial;
- on or before the second trial;
- after the third trial.

In a particular country, 18% of adults wear contact lenses. Adults are randomly selected and interviewed one at a time. Find the probability that the first adult who wears contact lenses is:
- one of the first 15 interviewed;
- not one of the first nine interviewed.

A coin is biased such that the probability of obtaining heads with each toss is equal to 1/5 . The coin is tossed until the first head is obtained. Find the probability that the coin is tossed:
- at least six times;
- fewer than eight times.

Exercise

Answers

Mode of the geometric distribution

With all geometric distributions:
- X=1 is the most likely value; and
- P(X=r) decreases as r increases (because 1-p is less than 1 and is being multiplied each time).

Expectation of the geometric distribution

If $X \sim geo(P)$ , then E(X) = 1/p.

Worked Example 1

One in four boxes of cereal contains a free gift. Let the random variable X be the number of boxes that a child opens, up to and including the one in which they find their first gift.