Question

24.3% of flowers of a certain species bloom "early" (before May 1st). You work for an arboretum and have a display of these flowers.

a) In a row of 40 flowers, what is the probability that 11 will bloom early?

b) In a row of 40 flowers, what is the probability that fewer than 12 will bloom early?

c) 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.)

d) In a row of 30 flowers, what is the probability that more than 10 will bloom early?

e) In a row of 30 flowers, what is the probability that between 5 and 10 (inclusive) will bloom early?

f) In a row of flowers, what is the probability that you will have to observe 5 flowers in order to see the first one that blooms early?

g) In a row of flowers, what is the probability that you will observe more than 5 flowers to see the first one that blooms early?

Answer 1

In a row of 40 flowers, we can find the probability of a specific number of flowers blooming early using the binomial distribution formula.

Step-by-step explanation:

a) Probability that 11 flowers will bloom early:

The probability of a flower blooming early is 24.3%. To calculate the probability of exactly 11 out of 40 flowers blooming early, we can use the binomial probability formula. The formula is: P(X=k) = (n C k) * (p^k) * (1-p)^(n-k), where n = number of trials (40), k = number of successful trials (11), p = probability of success (0.243), (n C k) = combination (n choose k).

Using the formula, we can calculate:

P(X=11) = (40 C 11) * (0.243^11) * (0.757^29)

b) Probability that fewer than 12 flowers will bloom early:

To find the probability of fewer than 12 flowers blooming early, we can sum up the probabilities of 0 to 11 flowers blooming early. We can use the binomial distribution formula for each value, and then add them all up.

P(X < 12) = P(X=0) + P(X=1) + P(X=2) + ... + P(X=11)

c) The expected number of flowers to observe to see the first one that blooms early:

The average number of flowers you need to observe to see the first one that blooms early can be calculated using the expected value formula. The expected value is calculated by multiplying the probability of an event by the number of trials it takes for that event to occur. In this case, the event is seeing the first flower that blooms early.

Expected value = 1 / p = 1 / 0.243 = 4.11

d) Probability that more than 10 flowers will bloom early:

To find the probability of more than 10 flowers blooming early, we can subtract the probability of 0 to 10 flowers from 1 (total probability).

P(X > 10) = 1 - (P(X=0) + P(X=1) + P(X=2) + ... + P(X=10))

e) Probability that between 5 and 10 flowers will bloom early:

To find the probability that between 5 and 10 flowers (inclusive) will bloom early, we can sum up the probabilities of 5 to 10 flowers blooming early. We can use the binomial distribution formula for each value, and then add them all up.

P(5 ≤ X ≤ 10) = P(X=5) + P(X=6) + P(X=7) + P(X=8) + P(X=9) + P(X=10)

f) Probability that you will have to observe 5 flowers to see the first one that blooms early:

The probability of seeing the first flower that blooms early after observing 5 flowers can be calculated using the binomial distribution formula.

P(X=1) = (5 C 1) * (0.243^1) * (0.757^4)

g) Probability that you will observe more than 5 flowers to see the first one that blooms early:

The probability of observing more than 5 flowers to see the first one that blooms early can be calculated by summing the probabilities of observing 6 or more flowers.

P(X > 5) = P(X=6) + P(X=7) + P(X=8) + ...