A binomial experiment is one that possesses the following properties:
The experiment consists of n repeated trials;
Each trial results in an outcome that may be classified as a success or a failure (hence the name, binomial);
The probability of a success, denoted by p, remains constant from trial to trial and repeated trials are independent.
The number of successes X in n trials of a binomial experiment is called a binomial random variable.
The probability distribution of the random variable X is called a binomial distribution.
n = the number of trials
x = 0, 1, 2, ... n [sample space]
p = the probability of success in a single trial
q = the probability of failure in a single trial
q= 1 −p
Mean and Variance of Binomial Distribution
If p is the probability of success and q is the probability of failure in a binomial trial, then the expected number of successes in n trials (i.e. the mean value of the binomial distribution) is
E(X) = μ = np
The variance of the binomial distribution is
V(X) = σ2 = npq
In a binomial distribution, only 2 parameters, namely n and p, are needed to determine the probability.
EXAMPLES
A die is tossed 3 times. What is the probability of no fives turning up?
A die is tossed 3 times. What is the probability of 1 five turning up?
A die is tossed 3 times. What is the probability of 3 fives ?
Hospital records show that of patients suffering from a certain disease, 75% die of it. What is the probability that of 6 randomly selected patients, 4 will recover?
This is a binomial distribution because there are only 2 outcomes (the patient dies, or does not).
X =number who recover
n = 6 ; x=4
p= 0.25 probability of success
q=0.75 probability of failure
Let's make a table and graph first.
Insert a Scatter Chart, then change the siries type to Bar Chart
Hospital records show that of patients suffering from a certain disease, 75% die of it. What is the probability that of 6 randomly selected patients, 4 will recover?
P(x=4)=3.29%
A Japanese archer finds that on the average he hits the middle of a target 4 times out of 5. If he fires 4 shots, what is the probability of
(A) more than 2 hits?
n= 4 p=4/5
P(X>2)=P(x=3)+P(x=4)
(B) at least 3 misses?
3 misses =1 hit
4 misses = 0 hits.
A manufacturer of metal pistons finds that on the average, 12% of his pistons are rejected because they are either oversize or undersize. What is the probability that a batch of 10 pistons will contain
(A) no more than 2 rejects?
In this case, "success" means rejection.
n=10 p=0.12
P(X<=2)=P(X=0)+P(X=1)+P(X=2)
(B) at least 2 rejects?
SOLUTION 1:
SOLUTION 2:
=1−P(X≤1)
=1−(P(x0)+P(x1))
=1−(0.2785+0.37977)
=0.34173
The ratio of boys to girls at birth in Singapore is at 1.09 : 1.
What proportion of Singapore families with exactly 6 children will have at least 3 boys? (Ignore the probability of multiple births.)
A manufacture is making a product with 20% defect rate. If we select 5 items at the end of the assembly line, what is the probability of having 1 defected item?
C(5,1)=5 WE GOT 5 DIFFRENCE WAYS HOW TO DO THAT.
