Question

For the binomial distribution with n = 10 and p = 0.3, find the probability of: 1. Five or more successes. 2. At most two successes. 3. At least one success. 4. At least 50% successes

Answer #1

Given in the question

No. of sample = 10

Probability of success P(Success) = 0.3

binomial distribution formula is

P(X=x | n) = nCx*(p)^x * (1-p)^(n-x)

Solution(1)

P(X>=5) = P(X=5) + P(X=6) + P(X=7) + P(X=8) + P(X=9) + P(X=10) =
10C5*(0.3)^5*(0.7)^5 +10C6*(0.3)^6*(0.7)^4 +10C7*(0.3)^7*(0.7)^3
+10C8*(0.3)^8*(0.7)^2 +10C9*(0.3)^9*(0.7)^1 +10C10*(0.3)^10*(0.7)^0
= 0.1503

Solution(2)

P(X<=2) = P(X=0) + P(X=1) + P(X=2) = 10C0*(0.3)^0*(0.7)^10
+10C1*(0.3)^1*(0.7)^9 +10C2*(0.3)^2*(0.7)^8 = 0.3828

Solution(3)

P(X>=1) = 1-P(X<1) = 1 - P(X=0) = 1- 10C0*(0.3)^0*(0.7)^10 =
1-0.0282 = 0.9718

Solution(4)

At least 50% Success means Success should be 5 or more

P(X>=5) = P(X=5) + P(X=6) + P(X=7) + P(X=8) + P(X=9) + P(X=10) =
10C5*(0.3)^5*(0.7)^5 +10C6*(0.3)^6*(0.7)^4 +10C7*(0.3)^7*(0.7)^3
+10C8*(0.3)^8*(0.7)^2 +10C9*(0.3)^9*(0.7)^1 +10C10*(0.3)^10*(0.7)^0
= 0.1503

Consider a binomial probability distribution with p=.35 and n=7.
what is the probability of the following?
a. exactly three successes P(x=3)
b. less than three successes P(x<3)
c. five or more successes P(x>=5)

Assume that Y is distributed according to a binomial
distribution with n trials and probability p of success.
Let g(p) be the probability of obtaining either no successes or all
successes, out of n trials. Find the MLE
of g(p).

Assume that a procedure yields a binomial distribution with with
n=8 trials and a probability of success of p=0.90. Use a binomial
probability table to find the probability that the number of
successes x is exactly 4.
1. P(4)= ?

Using your knowledge of Binomial Distribution find:
a. P(5) if p = 0.3, n = 7 Answer with at least 5 decinal
places.
b. P(x < 3) if p = 0.65, n = 5
c. P(x > 4) if p = 0.25, n = 6
d. P(at least one) if p = 0.45, n = 6
e. (A-Grade) P(x ≤ 17) if p = 0.75, n = 20

approximate the following binomial probabilities for
this continuous probability distribution
p(x=18,n=50,p=0.3)
p(x>15,n=50,p=0.3)
p(x>12,n=50,p=0.3)
p(12<x<18,n=50,p,=0.3)

If you are conducting a binomial experiment with p = 0.3 and n =
10:1) What is the mean AND standard deviation of successes?
2) What is P(x = 0)?
3) What is P(x = 1)?
4) What is P(x = 2)?
5) What is P(x ≤ 2)?

Binomial Distribution: 12:17
Notes:
the binomial distribution is a discrete probability
distribution
the parameters of binomial distribution are p (probability of
success in a single trial) and n (number of trials)
the mean of binomial distribution is np
the standard deviation of binomial distribution is sqrt(npq)
q=1-p
1. In the production of bearings, it is found out that 3% of
them are defective. In a randomly collected sample
of 10 bearings, what is the probability of
Given: p = 0.03,...

Study the binomial distribution table. Notice that the
probability of success on a single trial p ranges from
0.01 to 0.95. Some binomial distribution tables stop at 0.50
because of the symmetry in the table. Let's look for that symmetry.
Consider the section of the table for which n = 5. Look at
the numbers in the columns headed by p = 0.30 and
p = 0.70. Do you detect any similarities? Consider the
following probabilities for a binomial experiment...

(Binomial Test) N = 10, and we obtained 6 successes and 4
failures. What is the one-tailed probability of 6 or more successes
when the probability of a success is:
a. .50
b. .40
c. .30
d. .20
e. .10
For each of those probabilities, with N = 10, what is the mean
and standard deviation of the distribution?
a. when p = .50
b. when p = .40
c. when p = .30
d. when p = .20
e....

Assume that a procedure yields a binomial distribution with a
trial repeated times. Find the probability of successes given the
probability p of success on a given trial.
A. n = 12, x = 4, p = 0,40
B. n = 15, x = 2, p = 0.30
show all of your work

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