Question

35% of flowers of a certain species bloom "early" (before May 1st). You work for an arboretum and have a display of these flowers. Round all answers to three decimal places.

a) In a row of 30 flowers, what is the probability that 11 will bloom early?

b) As you walk down a row of these flowers, how many flowers do you expect to have to observe (on average) in order to see the first one that blooms early? (Keep your answer as a decimal.)

c) In a row of flowers, what is the probability that you will have to observe 4 flowers in order to see the first one that blooms early?

d) In a row of flowers, what is the probability that you will observe more than 4 flowers to see the first one that blooms early?

e) In a row of flowers, what is the probability that you will observe 4 flowers to see the 3 that bloom early?

Answer 1

35% of flowers of a certain species bloom "early" (before May 1st) This is same as the probability that a randomly selected flower will boom early is 0.35

a) In a row of 30 flowers, what is the probability that 11 will bloom early?

Let X be the number of flowers out of 30 that bloom early. X has a Binomial distribution with parameters, number of trials (number of flowers in the row) n=30, and success probability ( the probability that a randomly selected flower will boom early) p=0.35

The probability that X=x will bloom early is

the probability that 11 will bloom early is

ans: the probability that 11 will bloom early is 0.147

b) As you walk down a row of these flowers, how many flowers do you expect to have to observe (on average) in order to see the first one that blooms early? (Keep your answer as a decimal.)

Let X be the number of flowers that one has to observe in order to see the first one that blooms early. We can say that X has a Geometric distribution with parameter, success probability (the probability that a randomly selected flower will boom early) p=0.35

The expected value of X, using the formula for Geometric distribution, is

ans: you expect to have to observe (on average) 2.857 flowers in order to see the first one that blooms early

c) In a row of flowers, what is the probability that you will have to observe 4 flowers in order to see the first one that blooms early?

Let X be the number of flowers that one has to observe in order to see the first one that blooms early. We can say that X has a Geometric distribution with parameter, success probability (the probability that a randomly selected flower will boom early) p=0.35

The probability that one has to observe X=x flowers order to see the first one that blooms early is

the probability that you will have to observe 4 flowers in order to see the first one that blooms early is

ans: the probability that you will have to observe 4 flowers in order to see the first one that blooms early is 0.096

d) In a row of flowers, what is the probability that you will observe more than 4 flowers to see the first one that blooms early?

Let X be the number of flowers that one has to observe in order to see the first one that blooms early. We can say that X has a Geometric distribution with parameter, success probability (the probability that a randomly selected flower will boom early) p=0.35

The probability that one has to observe X=x flowers order to see the first one that blooms early is

the probability that you will observe more than 4 flowers to see the first one that blooms early is

ans: the probability that you will observe more than 4 flowers to see the first one that blooms early is 0.179

e) In a row of flowers, what is the probability that you will observe 4 flowers to see the 3 that bloom early?

Let X be the number of flowers that one has to observe in order to see the 3 that blooms early. We can say that X has a Negative Binomial distribution with parameters, number of successes (number of flowers that bloom early) r=3, success probability (the probability that a randomly selected flower will boom early) p=0.35

The probability that one has to observe X=x flowers order to see the 3 that blooms early is

the probability that you will observe 4 flowers to see the 3 that bloom early is

ans: the probability that you will observe 4 flowers to see the 3 that bloom early is 0.084