Probability word problem

Question

A high school baseball player has a 0.176 batting average. In one game, he gets 4 at bats

a. What is the probability he will get exactly 2 hits in the game?

b. What is the probability he will get less than 2 hits in the game?

Answer 1

.176 x 4 = .704 hits on average for 4 at bats

he'll probably get 1 hit

2 hits: either HHNN, HNHN, HNNH, NHHN, NHNH, or NNHH, 6 ways to get 2 hits. each has probability (.176)^2(.824)^2 = (.030976)(.678976) =.02103196. Multiply that by 6 to get:

0.126191763 = 12.6191763% chance

= 12.62% chance of 2 hits

less than 2 hits: either 1 hit or no hits

1 hit: HNNN, NHNN, NNHN or NNNH

4 ways to get 1 hit, each with probability (.176)(.824)^3 = .0985

multiply by 4 = .3939 = 39.39% chance of 1 hit

just one way to get 0 hits: NNNN. (.824)^4 = 46.10%

add 39.39+ 46.10 = 85.49% chance of less than 2 hits

greater than 2 hits: 2 ways. 3 hits or 4 hits

4 hits: (.176)^4 = 9.595 x 10^-4 = 0.001 = 0.1% chance

3 hits: either HHHN, HHNH, HNHH or NHHH, 4 ways each with

(.824)(.176)^3 = 4.5 x 10^-3 = 0.0045 = 0.45% x 4 = 1.8% chance of 3 hits

add them

1.8+.1 = 1.9%

1.9 + 85.5 +12.6 = 100%

Pr(>2)+Pr(<2)+P(2) = 1