Please help with these Algebra 2 questions?

Question

1. If you guess the numbers at random, what is the probability of getting at least three answers correct on a five question true-false quiz. (Please show your work)

2. A caterer knows that, on average, there will be at least one broken egg in every 3 cartons. A carton contains 12 eggs. The caterer plans to serve 1200 eggs for breakfast. What is the best estimate for the number of cartons the caterer should buy :

a) 97 cartons

b) 100 cartons

c) 103 cartons

d) 112 cartons

(please explain (its either c or d)

Answer 1

1 is solved by binomial theory, and you can use Pascal's triangle to do it.  Use the row that starts 1 5 since there are 5 questions.  That row is

1 5 10 10 5 1 totaling 32

so 1/32 = prob all right

5/32 = prob 4 right

10/32 = prob 3 right

etc

so at least 3 is 3, 4, or 5; 1/32 + 5/32 + 10/32 = 16/32 or 1/2

On 2 if there were no broken eggs it would just be 1200/12 = 100 cartons.  But those 100 cartons contain at least 100/3 or 33 1/3 broken eggs so he should buy 36 extra, 3 extra cartons, 103 total.