Question

Consider the random variable y = the number of broken eggs in a randomly selected carton of one dozen eggs. Suppose the probability distribution of y is as follows.

y	0	1	2	3	4
p(y)	0.66	0.19	0.11	0.03	?

(a)

Only y values of 0, 1, 2, 3, and 4 have probabilities greater than 0. What is p(4)? (Hint: Consider the properties of a discrete probability distribution.)

(b)

How would you interpret p(1) = 0.19?

If you check a large number of cartons, the proportion that will have at most one broken egg will equal 0.19.The proportion of eggs that will be broken in each carton from this population is 0.19.    In the long run, the proportion of cartons that have exactly one broken egg will equal 0.19.The probability of one randomly chosen carton having broken eggs in it is 0.19.

(c)

Calculate P(y ≤ 2), the probability that the carton contains at most two broken eggs.

Interpret this probability.

If you check a large number of cartons, the proportion that will have at most two broken eggs will equal 0.96.The probability of two randomly chosen cartons having broken eggs in them is 0.96.    In the long run, the proportion of cartons that have exactly two broken eggs will equal 0.96.The proportion of eggs that will be broken in any two cartons from this population is 0.96.

(d)

Calculate P(y < 2), the probability that the carton contains fewer than two broken eggs.

Why is this smaller than the probability in part (c)?

This probability is less than the probability in Part (c) because the proportion of eggs with any exact number of broken eggs is negligible.This probability is not less than the probability in Part (c) because the two probabilities are the same for this distribution.    This probability is less than the probability in Part (c) because in probability distributions, P(y ≤ k) is always greater than P(y < k).This probability is less than the probability in Part (c) because the event y = 2 is now not included.

(e)

What is the probability that the carton contains exactly 10 unbroken eggs? (Hint: What is the corresponding value of y?)

(f)

What is the probability that at least 10 eggs are unbroken?

Answer 1