1. The Binomial Distribution
Thursday, 06 December 2018

2. X ~ Bin (n=4, p= 1/6) Note: q = 5/6 P(X = 0) P(X = 1) Example
A die is rolled four times and the number of sixes is recorded.
Work out the probability of obtaining 0, 1, 2, 3, or 4 sixes.
X ~ Bin (n=4, p= 1/6)
Note: q = 5/6
P(X = 0)
P(X = 1)

3. P(X = 2) =
= (4 dp)
P(X = 3)
P(X = 4)

4. Using either Pascal’s triangle or nCr
The above probabilities can all be obtained from the binomial expansion of
Using either Pascal’s triangle or nCr
P(X=4)
P(X=3)
P(X=2)
P(X=1)
P(X=0)
Hence this model is known as the Binomial Distribution

5. Characteristics of a Binomial Distribution
A binomial distribution consists of a fixed number of trials (n) where:
Each trial has only two outcomes, SUCCESS (p) and FAILURE (q).
The probability of failure, q = 1 – p.
The outcome of successive trials is independent.
X is the random variable such that X is the number of successes.

6. Example
A teacher has a box which contains thirty pencils. With students borrowing pencils the probability that a pencil needs sharpening is 0.2
Find the probability that
No pencil needs sharpening
Exactly 5 pencils need sharpening
No more than two need sharpening

7. X ~ Bin (n=30, p= 0.2) Note: q = 0.8 a) P(X = 0) b) P(X = 5)
c) P(X  2) = P(X = 0) + P(X = 1) + P(X = 2)

8. X = Number of people that recovered X ~ Bin (n=20, p = 0.4)
EXAMPLE
The probability that patient recovers from a disease is 0.4. If 20 people contracted the disease, what is the probability that:
13 people exactly recovered.
At least 18 people recovered.
X = Number of people that recovered X ~ Bin (n=20, p = 0.4)
P(X = 13) =
= (4 dp)

9. b) P(X ≥ 18) =
P(X = 18) + P(X = 19) + P(X = 20)

10. Example
The proportion of left-handed people in the world is approximately 13%.
Based on this proportion, find the probability that in a class of 25 pupils
There are no left-handed people
There are exactly three left-handed people
There is more than one left-handed person.