University Statistics Test Questions on Applying Binomial Distribution

📚 Quick Study Guide

• 🔢 Definition: The binomial distribution models the probability of obtaining exactly $k$ successes in $n$ independent Bernoulli trials, where each trial has the same probability $p$ of success.
• 🧪 Formula: The probability mass function (PMF) is given by: $P(X = k) = {n \choose k} * p^k * (1-p)^{(n-k)}$, where ${n \choose k} = \frac{n!}{k!(n-k)!}$ is the binomial coefficient.
• 📊 Assumptions:
  • 📈 Fixed number of trials ($n$).
  • ✅ Each trial is independent.
  • 🎲 Two possible outcomes (success or failure).
  • 🍀 Constant probability of success ($p$) for each trial.
• 📈 Mean: The expected value (mean) of a binomial distribution is $\mu = np$.
• 📊 Variance: The variance is $\sigma^2 = np(1-p)$.
• 💡 Standard Deviation: The standard deviation is $\sigma = \sqrt{np(1-p)}$.

Practice Quiz

1. A fair coin is tossed 5 times. What is the probability of getting exactly 3 heads?

   1. 0.250
   2. 0.3125
   3. 0.500
   4. 0.625

2. A multiple-choice test has 20 questions. Each question has 4 options, with only one correct answer. If a student randomly guesses the answer to each question, what is the probability that they will answer exactly 5 questions correctly?

   1. 0.2023
   2. 0.1561
   3. 0.0000
   4. 0.1029

3. It is known that 30% of the products from a manufacturing process are defective. If 10 products are randomly selected, what is the probability that exactly 2 of them are defective?

   1. 0.2335
   2. 0.3000
   3. 0.0000
   4. 0.2668

4. A basketball player makes 70% of their free throws. If they attempt 8 free throws in a game, what is the probability they make exactly 6?

   1. 0.2965
   2. 0.2541
   3. 0.5282
   4. 0.0808

5. A certain medication is effective in 80% of patients. If 12 patients are given the medication, what is the probability that it will be effective for at least 10 patients?

   1. 0.2835
   2. 0.5580
   3. 0.8000
   4. 0.9313

6. Suppose a die is tossed 9 times. What is the probability of getting a '3' exactly twice?

   1. 0.2791
   2. 0.2341
   3. 0.1000
   4. 0.3021

7. If $X \sim B(15, 0.4)$, find $P(X = 6)$.

   1. 0.2066
   2. 0.0000
   3. 0.4000
   4. 0.6000