Let y denote the number of broken eggs in a randomly selected
carton of one dozen “store brand” eggs at a certain market. Suppose
that the probability distribution of y is as follows.

y 0 1 2 3 4
p(y) .63 .22 .10 .04 ?

(a) Only y values of 0, 1, 2, 3, and 4 have positive
probabilities. What is p(4)?

(b) How would you interpret p(1) = .22?
The proportion of eggs that will be broken in each carton from this
population is .22.The probability of one randomly chosen carton
having broken eggs in it is .22.     In
the long run, the proportion of cartons that have exactly one
broken egg will equal .22.If you check a large number of cartons,
the proportion that will have at most one broken egg will equal
.22.

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

Interpret this probability.
The proportion of eggs that will be broken in any two cartons from
this population is .0.95.In the long run, the proportion of cartons
that have exactly two broken eggs will equal
.0.95.     If you check a large number of
cartons, the proportion that will have at most two broken eggs will
equal .0.95.The probability of two randomly chosen cartons having
broken eggs in them is .0.95.

(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
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 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 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?