# 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, nCk, 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

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

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:

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

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.

• Find the mode and the expectation of X; and
• Interpret the two values in the context of this question.

Worked Example 2

The random variable X follows a geometric distribution. Given that E(X) = 3½, find P(X>6).

Worked Example 3

Given that $X \sim geo(P)$ and that P(X≤3) = 819/1331, find:

• P(X>3)
• P(1<X≤3).

Exercise

Mixed Exercise