Class: STAT 131 Subject: probability Date: 2025-01-29 Teacher: **Prof. Marcela

Binomial Distribution

Formula

$P(X=K)=\left(\genfrac{}{}{0ex}{}{n}{k}\right)p^k(1-p{)}^{n-k}$ n = # of independent trials p = probability of success k = target value

Example

• It is known that 20% of the students on campus don’t have a TikTok account. If we collect a sample of $n=100$ students, what is the probability of observing exactly 40 students that don’t have a TikTok account?
• Also what is the probability of observing 30 students?

Explanation

1. Do we need assumptions? Yes

   • $n$ trials
   • independent
   • fixed $p$ (success: not having TikTok account, failure: having a TikTok account)

2. Identify the parameter values

   • $n$ = 100
   • $p$ = 0.2

3. What is the probability of interest? Formula: $P(X=K)=\left(\genfrac{}{}{0ex}{}{n}{k}\right)p^k(1-p{)}^{n-k}$

   • P(X = 40) = $\left(\genfrac{}{}{0ex}{}{100}{40}\right)(0.2{)}^{40}(0.8{)}^{60}=2.316e^{-}6$
   • P(x >= 30) =