Question

It has been reported that 42% of college students graduate in 4 years. Consider a random sample of thirty students, and let the random variable X be the number who graduate in 4 years.

Find the probability that 14 or fewer students in the sample graduate in 4 years.

Answer 1

This problem is related to binomial distribution. We will use the exact binomial x~bin(400,0.00825)

The below formulas are used to estimate the number of success and failures in n independent number of trials or experiments

Here, P(x) is the probability of x successes occur in the n number of events, p is the probability of success and q is the probability of failure often denoted by q = (1 - p).

We have find P(X < 14)

P(X < 14) = P(0) + P(1) + P(2) + P(3) + P(4) + P(5) + P(6) + P(7) + P(8) + P(9) + P(10) + P(11) + P(12) + P(13) + P(14)

In this case, n = 30, p = 0.42, q = 1 - 0.42 = 0.58, and x = 14

P(14) = ₃₀C₁₄*(0.42)¹⁴*(0.58)^(30-14) = (30!/14!*16!)*(0.42)¹⁴*(0.58)¹⁶ = 0.1267565

P(13) = ₃₀C₁₃*(0.42)¹³*(0.58)^(30-13) = (30!/13!*17!)*(0.42)¹³*(0.58)¹⁷ = 0.1441544

P(12) = ₃₀C₁₂*(0.42)¹²*(0.58)^(30-12) = (30!/12!*18!)*(0.42)¹²*(0.58)¹⁸ = 0.1437730

P(11) = ₃₀C₁₁*(0.42)¹¹*(0.58)^(30-11) = (30!/11!*19!)*(0.42)¹¹*(0.58)¹⁹ = 0.1253960

P(10) = ₃₀C₁₀*(0.42)¹⁰*(0.58)^(30-10) = (30!/10!*20!)*(0.42)¹⁰*(0.58)²⁰ = 0.0952413

P(9) = ₃₀C₉*(0.42)⁹*(0.58)^(30-9) = (30!/9!*21!)*(0.42)⁹*(0.58)²¹ = 0.0626303

P(8) = ₃₀C₈*(0.42)⁸*(0.58)^(30-8) = (30!/8!*22!)*(0.42)⁸*(0.58)²² = 0.0353821

P(7) = (30!/7!*23!)*(0.42)⁷*(0.58)²³ = 0.0169951

P(6) = (30!/6!*24!)*(0.42)⁶*(0.58)²⁴ = 0.0068453

P(5) = (30!/5!*25!)*(0.42)⁵*(0.58)²⁵ = 0.0022687

P(4) = (30!/4!*26!)*(0.42)⁴*(0.58)²⁶ = 0.0006025

P(3) = (30!/3!*27!)*(0.42)³*(0.58)²⁷ = 0.0001233

P(2) = (30!/2!*28!)*(0.42)²*(0.58)²⁸ = 0.0000182

P(1) = (30!/1!*29!)*(0.42)¹*(0.58)²⁹ = 0.0000017

P(0) = (30!/0!*30!)*(0.42)⁰*(0.58)³⁰ = 0.0000001

P(X < 14) = 0.1267565 + 0.1441544 + 0.1437730 + 0.1253960 + 0.0952413 + 0.0626303 + 0.0353821 + 0.0169951 + 0.0068453 + 0.0022687 + 0.0006025 + 0.0001233 + 0.0000182 + 0.0000017 + 0.0000001

P(X < 14) = 0.7601885