Question

There are 8 students. Each student has a fair quarter coin (that means the coin has two sides, head and tail, and the probability of getting a head is 0.5). Another student, whose name is Emily, asks the 8 students to flip their coins at the same time and counts the total number of heads she sees.

(1) What is the probability that Emily sees 3 heads?

a) 0.2188 b) 0.7812 c) 0.0039 d) 0.1250

(2) What is the probability that Emily sees 6 or less heads?

a) 0.8906 b) 0.0352 c) 0.1094 d) 0.9648

(3) What is the probability that Emily sees 6 or less tails?

a) 0.9648 b) 0.1445 c) 0.8555 d) 0.0352

(4) What is the probability that Emily sees between 2 (inclusive) and 6 (inclusive) heads?

a) 0.2734 b) 0.1797 c) 0.9298 d) 0.8203

Answer 1

1) Option A is correct.

P(X=3) = 0.2188

2) Option D is correct.

P(X ≤ 6) = 0.9648

3) Option A is correct.

P(X ≥ 2) = 0.9648

4) Option C is correct.

P(2 ≤ X ≤ 6) = 0.9298

Explanation:

This is a binomial distribution problem

Binomial distribution function is represented by

P(X = x) = ⁿCₓ pˣ qⁿ⁻ˣ

n = total number of sample spaces = 8 students with the coins

x = Number of successes required = variable

p = probability of success = probability of a head = 0.5

q = probability of failure = probability of a tail = 1 - 0.5 = 0.5

a) Probability that Emily sees exactly 3 heads

P(X=3)

x = 3

P(X=3) = ⁸C₃ (0.5)³ (0.5)⁸⁻³

P(X=3) = 0.21875 = 0.2188

2) Probability that Emily sees 6 or less heads

P(X ≤ 6)

P(X ≤ 6) = P(X=0) + P(X=1) + P(X=2) + P(X=3) + P(X=4) + P(X=5) + P(X=6)

Solving each of these and adding the answers

P(X ≤ 6) = 0.96484375 = 0.9648

3) Probability that Emily sees 6 or less tails = Probability that Emily sees 2 or more heads

P(X ≥ 2) = P(X=2) + P(X=3) + P(X=4) + P(X=5) + P(X=6) + P(X=7) + P(X=8)

Solving each of these and summing them all up

P(X ≥ 2) = 0.96484375 = 0.9648

4) Probability that Emily sees between 2 (inclusive) and 6 (inclusive) heads

P(2 ≤ X ≤ 6) = P(X=2) + P(X=3) + P(X=4) + P(X=5) + P(X=6)

Solving each of these and summing them all up

P(2 ≤ X ≤ 6) = 0.109375 + 0.21875 + 0.2734375 + 0.21875 + 0.109375 = 0.9296875 = 0.9297 ≈ 0.9298

Hope this Helps!!!