For these binomial distribution probability problems you may use your calculator
ID: 3337396 • Letter: F
Question
For these binomial distribution probability problems you may use your calculator but you need to state what p, n, and x are along with the probability.
1) A manufacturer of halogen bulbs knows that 3% of the production of their 100 W bulbs will be defective. What is the probability that exactly 5 bulbs in a carton of 144 bulbs will be defective?
2) A fair die has four faces numbered one to four. What is the probability of rolling a two exactly three times in ten rolls of the die?
3) A packet of vegetable seeds has a germination rate of 96%. What is the probability that exactly 10 of 12 seeds planted will sprout?
4) A student take a five question multiple-choice quiz. Each question has four possible responses. The student guesses at random for each question. Calculate the probability for that if a student guesses they get exactly 4 correct.
5) There are 10 members on a committee. The probability of any member attending a randomly chosen meeting is 0.9. The committee cannot do business if more than 3 members are absent. What is the probability that 7 or more members will be present on a given date?
Explanation / Answer
Q1.
pmf of B.D is = f ( k ) = ( n k ) p^k * ( 1- p) ^ n-k
where
k = number of successes in trials
n = is the number of independent trials
p = probability of success on each trial
I.
mean = np
where
n = total number of repetitions experiment is excueted
p = success probability
mean = 144 * 0.03
= 4.32
II.
variance = npq
where
n = total number of repetitions experiment is excueted
p = success probability
q = failure probability
variance = 144 * 0.03 * 0.97
= 4.1904
III.
standard deviation = sqrt( variance ) = sqrt(4.1904)
=2.047
X ~ B(144, 0.03)
P( X = 5 ) = ( 144 5 ) * ( 0.03^5) * ( 1 - 0.03 )^139
= 0.1694
Q2.
X ~ B(10,1/4)
P( X = 3 ) = ( 10 3 ) * ( 0.25^3) * ( 1 - 0.25 )^7
= 0.2503
Q3.
X ~ B(12, 0.96)
P( X = 10 ) = ( 12 10 ) * ( 0.96^10) * ( 1 - 0.96 )^2
= 0.0702
Q4.
X ~ B(5, 0.25)
P( X = 4 ) = ( 5 4 ) * ( 0.25^4) * ( 1 - 0.25 )^1
= 0.0146