1. Binomial Distribution
GCSE Statistics
Binomial Distribution

2. A binomial distribution occurs when there are a fixed number of independent trials(n), each trial having only two outcomes.
Examples of this are heads or tails on a coin, 6 or not 6 on a die,
The outcomes are defined as success or failure.
The probability of success is p.
The probability of failure is q, where q = 1 – p (since p + q = 1)
The binomial distribution is written as B(n,p) where n is the number of trials and p is the probability of success.
If I flip a coin 20 times the binomial distribution is expressed as
B(20, 0.5) or B(20, ½)

3. The probabilities for the events of n binomial trials are the terms of the expansion (p + q)n
so if there are 2 trials we get (p + q)2 = p² + 2pq + q²
for 3 trials (p + q)3 = p³ + 3p²q + 3pq² + q³
for 4 trials (p + q)⁴ = p⁴ + 4p³q² + 6p²q² + 4pq³ + q⁴
In the exam you will be expected to deal with values of n up to 2, but will only be expected to remember the expansion for n = 2.

4. (p + q)⁴ = p⁴ + 4p³q + 6p²q² + 4pq³ + q⁴
For example, the probability of an oboe reed faulty is 0.1 and a packet holds four reeds. You want to find the probability there are one or fewer faulty oboe reeds. You are given that:
(p + q)⁴ = p⁴ + 4p³q + 6p²q² + 4pq³ + q⁴
(4 good) (3 good 1 faulty) (2 good 2 faulty) (1 good 3 faulty) (4 faulty)
in this case P(success) = 1 – 0.1 = q = P(failure) = 0.1
one or fewer faulty reeds are the terms
p⁴ + 4p³q² = 0.9⁴ + (4 x 0.9³ x 0.1) = =
the probability of having one or fewer faulty oboe reeds in a pack is (94.77%)

5. Your turn
Exercise 8c page 300