Statistics and Probability Answers

Given the Standard Normal distribution, find the following

(a) P(Z < 1.8)

(b) P(−1.1 < Z ≤ 1.8)

(c) P(−1.8 ≤ Z ≤−1.1)

(d) P(Z > −2.5)

(e) P(Z > −0.95)

(f) P(Z < −0.95)

(g) P(Z ≥ 2.18)

(h) P(Z > 10)

The number of claims per month paid by an insurance company is modelled by a random variable N with p.m.f satisfying the relation

p(n + 1) =

1/3p(n), n = 0,1,2,...

where p(n) is the probability that n claims are filed during a given month

(a) Find p(0).

(b) Calculate the probability of at least one claim during a particular month given that there have been at most four claims during the month. 

 The mean number of automobiles entering a mountain tunnel per two-minute period is one. An excessive number of cars entering the tunnel during a brief period of time produces a hazardous situation. Find the probability that the number of autos entering the tunnel during a two-minute period exceeds three. Does the Poisson model seem reasonable for this problem? 

(a) The random variable Y has a Poisson distribution and is such that P(Y = 0) = P(Y = 1). What is P(Y 2 = 1)?

(b) Cars arrive at a toll both according to a Poisson process with mean 80 cars per hour. If the attendant makes a one-minute phone call, what is the probability that at least 1 car arrives during the call?

 Customers arrive at a checkout counter in a department store according to a Poisson distribution at an average of seven per hour. During a given hour, what are the probabilities that

(a) no more than three customers arrive?

(b) at least two customers arrive?

(c) exactly four customers arrive? 

A salesperson has found that the probability of a sale on a single contact is approximately .03. If the salesperson contacts 100 prospects, what is the approximate probability of making at least one sale?

A shipment of 20 cameras includes 3 that are defective. What is the minimum number of cameras that must be selected if we require that P(at least 1 defective)≥ .8?

A warehouse contains ten printing machines, four of which are defective. A company selects five of the machines at random, thinking all are in working condition. What is the probability that all five of the machines are nondefective?

An urn contains ten marbles, of which five are green, two are blue, and three are red. Three marbles are to be drawn from the urn, one at a time without replacement. What is the probability that all three marbles drawn will be green?

Given that we have already tossed a balanced coin ten times and obtained zero heads, what is the probability that we must toss it at least two more times to obtain the first head?