# Notes on Binomial Theorem

```Find the mean &amp; St. Dev for each.
Mean
St. Dev
X
8
3
Y
12
4
4x ī­ 7
3x ī­ 2 y
Find the mean &amp; standard deviation
x
p(x)
4
0.2
7
0.31
12
0.24
15
0.18
18
0.07
BINOMIAL
DISTRIBUTION
FORMULA
Section 6.3A
Binomial Experiment - BINS
• Binary?
• There are two possible outcomes – success and failure
• Independent?
• Trials must be independent
• Number?
• The number of trials must be fixed – out of n….
• Success?
• On each trial the probability of success must be the same
Let E be the event that a person goes out shopping on Black Friday.
Find the probability distribution for a sample of 3 people if the P(E) = .78.
Probability distribution of going shopping on black
Friday for 4 people. P(E) = .78
Let’s Develop the formula!
• If I have 5 people, how many ways can I arrange them?
đâđâđâđâđ
• How many ways can I choose 2 of the 5 who went
shopping on black Friday?
[đ â đ] â đ â đ â đ
đ!
=
đâđâđ
đ−đ !
• But every set of two is the same (SSNNN = SSNNN)…
đ!
đ!âđ!
Binomial Distribution
īĻ n īļ x nī­ x
p( x) īŊ ī§ īˇ p q
ī¨ xī¸
īĻnīļ
n!
n Cx īŊ ī§ īˇ īŊ
ī¨ x ī¸ (n ī­ x)! x !
where
q īŊ 1ī­ p
The survival rate for a risky operation is 80%. What is the
probability that exactly 4 of the next 5 patients survive?
USA today stated that 45% of Americans know somebody
who became addicted to a drug other than alcohol. If we
randomly ask 15 people, what’s the probability that exactly 3
What’s the probability that at least 12 are
What the probability that less than 3 are