The Binomial Distribution A fair die is rolled eight times. Find the probability
ID: 3141775 • Letter: T
Question
The Binomial Distribution A fair die is rolled eight times. Find the probability that no more than 2 sixes come up. A large industrial firm allows a discount on any invoice that is paid within 30 days. Of all invoices, 10% receive the discount. In a company audit, 12 invoices are sampled at random. (a) What is the probability that fewer than 4 of the 12 sampled invoices receive the discount? (b) What is the probability that more than 1 of the 12 sampled invoices receives a discount? The Hypergeometric Distribution Of 50 buildings in an industrial park, 12 have electrical code violations. If 10 buildings are selected at random for inspection, what is the probability that exactly 3 of the 10 have code violations? A test of weld strength involves loading welded joints until a fracture occurs. For a certain type of weld, 80% of the fractures occur in the weld itself, while the other 20% occur in the beam. A number of welds are tested. a. Let X be the number of tests up to and including the third beam fracture. What is the distribution of X? Find P(X = 8). b. Geometric Distribution:Let x be the number of tests up to and including the first test that results in a beam failure. What is the distribution of X? Find P(X = 3). The Poisson Distribution Particles are suspended in a liquid medium at a concentration of 6 particles per ml. A large volume of the suspension is thoroughly agitated, and then 3 mL are withdrawn. What is the probability that exactly 15 particles are withdrawn? Grandma bakes chocolate chip cookies in batches of 100. She puts 300 chips into the dough. When the cookies are done, she gives you one. What is the probability that your cookie contains no chocolate chips? Assume that the number of hits on a certain website during a fixed time interval follows a Poisson distribution. Assume that the mean rate of hits is 5 per minute. Find the probability that there will be exactly 17 hits in the next three minutes.Explanation / Answer
Solution:
1.
We want to find the probability of rolling no more than 2 sixes. "no more than 2" can be rephrases as "2 or fewer" or "less than or equal to 2".
The possible results are:
0 sixes
1 six
2 sixes
This is a binomial distribution problem, since the two possible results are {6, not 6}, the probability of rolling a 6 doesn't change between rolls, and there's a fixed number of trials (8 rolls.)
With a probability of a single rolls success p, and failure q, the formula for finding the probability of x successes out of n rolls is:
P(x) = nCx px qn-x
The probability p of rolling a 6 in a single roll is 1/6.
The probability q of not rolling a 6 is 5/6.
n = 8.
You then need to calculate
P(x2) = P(x=0) + P(x=1) + P(x=2) +
P(x=0) = 8C0 (1/6)0 (5/6)8 = (5/6)8
P(x=1) = 8C1 (1/6)1 (5/6)7 = 8* (1/6) * (5/6)7
P(x=2) = 8C2 (1/6)2 (5/6)6 = 28 *(1/6)2(5/6)6
therefore
P(x2) = 0.232568 + 0.3721088 + 0.260476
P(x2) = 0.8652
Answer: 0.8652
Related Questions
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.