2. Chapter Goals
After completing this chapter, you should be
able to:
Identify probability problems with binomial distributions
Describe and compute probabilities for a Binomial
distributions
Identify probability problems with Poisson distribution
Describe and compute probabilities for a Poisson
distributions
Find mean and variance for Binomial and Poisson
Distributions
3. Chapter Goals
After completing this chapter, you should be
able to:
Identify probability problems with normal distribution
Describe the characteristics normal distributions
Translate normal distribution problems into standardized
normal distribution problems
Find probabilities using a normal distribution table
(continued)
6. Binomial Distribution
Consider an experiment with only two outcomes:
“success” or “failure” - The Bernoulli Trial
Let p denote the probability of success
Let q be the probability of failure and q =1 – p
If there are n number independent Bernoulli Trials
And let random variable X= no. of success
X has a Binomial R.V. i.e it has a Binomial p.d.f.
𝑋~𝐵(𝑛, 𝑝) 𝑛 is the no. of trials
𝑝 the prob. of success in each trial
7. Binomial Distribution Formula
𝑃 𝑋 = 𝑥 = 𝑛
𝐶𝑥𝑝𝑥
𝑞 𝑛−𝑥
𝑃(𝑋 = 𝑥) = probability of 𝑥 successes in 𝑛 trials,
with probability of success 𝑝 on each trial
𝑥= number of ‘successes’ in sample,
(𝑥 = 0, 1, 2, ..., 𝑛)
𝑛 = sample size (number of trials
or observations)
𝑝 = probability of “success”
Example: Flip a coin four
times, let x = # heads:
𝑛 = 4
𝑝= 0.5
1 - 𝑝= (1 - 0.5) = 0.5
x = 0, 1, 2, 3, 4
8. Sequences of x Successes
in n Trials
The number of sequences with x successes in n
independent trials is:
Where n! = n·(n – 1)·(n – 2)· . . . ·1 and 0! = 1
These sequences are mutually exclusive, since no two
can occur at the same time
x)!
(n
x!
n!
Cx
n
9. Example:
Calculating a Binomial Probability
What is the probability of one success in five
observations if the probability of success is 0.1?
x = 1, n = 5, and p = 0.1
.32805
.9)
(5)(0.1)(0
0.1)
(1
(0.1)
C
p)
(1
p
C
1)
P(x
4
1
5
1
1
5
x
n
x
x
n
10. Example:
Calculating a Binomial Probability
A fair dice is rolled six times. Let 𝑋 be the
number of times the outcome is a multiple of 3.
Find the probability distribution function of 𝑋
Success = getting no. multiple of 3 = 3 or 6
)
6
(
3292
.
0
3
2
3
1
)
2
(
2634
.
0
3
2
3
1
)
1
(
0878
.
0
3
2
3
1
)
0
(
6
,
3
2
,
3
1
6
2
p
4
2
2
6
5
1
1
6
6
0
0
6
X
P
sampai
buat
C
X
P
C
X
P
C
X
P
n
q
11. Example:
Calculating a Binomial Probability
A fair dice is rolled six times. Let 𝑋 be the
number of times the outcome is a multiple of 3.
Find the probability distribution function of 𝑋
𝑿 = 𝒙 0 1 2 3 4 5 6
𝑃(𝑋 = 𝑥) 0.0878 0.2634 0.3292 0.2195 0.0823 0.0165 0.0014
12. Using Binomial Tables
• We can use statistical table for Binomial
Probability distribution.
• However, the table gives the cumulative probability
for a given values of 𝑛 and 𝑝
• The table is given on page 307-312 in the text book
14. Example:
Calculating a Binomial Probability
Let 𝑋be a random variable with 𝑛=10 and 𝑝=0.4, that
is 𝑋~𝐵 10,0.4 . Find the following probabilities:
a) 𝑃(𝑋 ≥ 4) = 0.6177 (read direct from the table)
b) 𝑃(𝑋 < 2) = 1 − 𝑃(𝑋 ≥ 2)
= 1 − 0.9536 = 0.0464
c) 𝑃(𝑋 > 8) =𝑃(𝑋 ≥ 9)= 0.0017
d) 𝑃(2 ≤ 𝑋 ≤ 5) =𝑃 𝑋 ≥ 2 − P(X ≥ 6)
= 0.9536 − 0.1662 = 0.7874
e) 𝑃(𝑋 = 3) = 𝑃 𝑋 ≥ 3 − 𝑃(𝑋 ≥ 4) = 0.8327 − 0.6177
= 0.2150
15. Possible Binomial Distribution
Settings
A manufacturing plant labels items as
either defective or acceptable
A firm bidding for contracts will either get a
contract or not
A marketing research firm receives survey
responses of “yes I will buy” or “no I will
not”
New job applicants either accept the offer
or reject it
16. Binomial Distribution
The shape of the binomial distribution depends on the
values of p and n
n = 5 p = 0.1
n = 5 P = 0.5
Mean
0
.2
.4
.6
0 1 2 3 4 5
x
P(x)
.2
.4
.6
0 1 2 3 4 5
x
P(x)
0
Here, n = 5 and p = 0.1
Here, n = 5 and p = 0.5
17. Binomial Distribution
Mean and Variance
Mean
Variance and Standard Deviation
np
E(x)
μ
p)
-
np(1
npq
σ2
p)
-
np(1
σ
npq
Where n = sample size
p = probability of success
q=(1 – p) = probability of failure
