Download 16.3 Discrete Math Section 16.3

Binomial Probability Theorem
• In a rainy season, there is 60% chance that it will rain
on a particular day. What is the probability that there
will exactly 4 rainy days in a week?
Discrete Math Section 16.3
Use the Binomial Probability theorem to find the probability of a given
outcome on repeated independent trials.
Flip a coin three times - each flip is
independent of all the previous flips
Outcomes:
HHH
HHT
THH
THT
HTH
TTH
HTT
TTT
Groups
probability
1
3H
1/8
3 2H 1T
3/8
3 1H 2T
3/8
1
3T
1/8
Pascal’s Triangle
1
1
1
1
2
1
1
3
3
1
Binomial Probability Theorem
Suppose an experiment consists of a sequence
of n repeated independent trials, each trial
having two possible outcomes, A or not A. If
on each trial P(A) = p and P(not A) = 1-p, then
the binomial expansion [ p + (1-p)]n gives the
probabilities for all occurrences of A.
n+ C
n-1 (1-p)1 + C
n-2 (1-p)2 … C (1-p)n
C
p
p
p
n n
n n-1
n n-2
n 0
Examples
A coin is tossed 8 times. What is the probability of
getting exactly 5 heads and 3 tails?
note: two outcomes…heads or tails
repeated independent trials
P(H) = p = ½
P(T) = 1-p = ½
8C5
p5 (1-p)3
5 (1/2)3 = 7/32
C
(1/2)
8 5
example
A quiz consists of 10 multiple choice questions.
Each question has three possible answers. If
you guess at each question, what is the
probability of getting at least eight correct
answers?
example
A die is rolled four times. What is the
probability of getting exactly three fives?
Assignment
Page 616
Problems 2-12 even, 15,17