probability questions

Question

What is the probability of rolling a number less than 3 at least 3 times in 5 rolls of a six sided die?

Answer 1

Prob of <3 is 1/3. Call this x

Prob of 3 or more 2/3 Call this y

Interested iwhen <3 occurs 3,4 or five times in five rolls

Binomial expansion of (x+y)^5 is

x^5 + 5x^4y +10x^3y^2 +10x^2y^3 +5xy^4 +y^5

We are interested in the first 3 terms

(1/3)^5 +5(1/3)^4(2/3) +10 (1/3)^3(2/3)^2= 1/243 +5(1/81)(2/3) + 10(1/3)^3(2/3)^2=

1/243 + 10/243 + 40/243 = 51/243 = .21