Q1: A binomial random variable is characterized by which of the following conditions? (a) Each trial has exactly two possible outcomes (b) The probability of success changes from trial to trial (c) The number of trials is infinite (d) The trials are dependent on each other
Solution:
Ans: (a) Explanation: A binomial random variable requires that each trial has exactly two possible outcomes (success or failure), the trials are independent, the number of trials \(n\) is fixed, and the probability of success \(p\) remains constant for each trial. Option (b) is incorrect because \(p\) must be constant. Option (c) is incorrect because \(n\) must be finite. Option (d) is incorrect because trials must be independent.
Q2: If \(X\) is a binomial random variable with \(n = 10\) and \(p = 0.3\), what is the probability of exactly 3 successes? Use \(P(X = k) = \binom{n}{k}p^k(1-p)^{n-k}\). (a) 0.2668 (b) 0.3000 (c) 0.1500 (d) 0.4500
Solution:
Ans: (a) Explanation: Using the binomial probability formula: \(P(X = 3) = \binom{10}{3}(0.3)^3(0.7)^7\) \(\binom{10}{3} = \frac{10!}{3!7!} = 120\) \(P(X = 3) = 120 \times 0.027 \times 0.0824 \approx 0.2668\) Option (b) would result from incorrectly using just \(p\), option (c) is too small, and option (d) is unrealistically large for this scenario.
Q3: The mean of a binomial distribution with \(n = 20\) and \(p = 0.4\) is: (a) 8 (b) 12 (c) 4.8 (d) 6
Solution:
Ans: (a) Explanation: The mean (expected value) of a binomial distribution is calculated using the formula \(\mu = np\). \(\mu = 20 \times 0.4 = 8\) Option (b) results from using \(p = 0.6\), option (c) results from calculating the variance instead, and option (d) is simply incorrect.
Q4: What is the variance of a binomial random variable with \(n = 15\) and \(p = 0.6\)? (a) 9 (b) 3.6 (c) 5.4 (d) 2.32
Solution:
Ans: (b) Explanation: The variance of a binomial distribution is given by \(\sigma^2 = np(1-p)\). \(\sigma^2 = 15 \times 0.6 \times 0.4 = 15 \times 0.24 = 3.6\) Option (a) is the mean, not the variance. Option (c) results from using \(np \times 0.6\). Option (d) is the standard deviation squared incorrectly.
Q5: A student takes a 10-question true/false quiz and guesses randomly on each question. What is the probability the student gets exactly 5 questions correct? (a) 0.2461 (b) 0.5000 (c) 0.1250 (d) 0.3750
Solution:
Ans: (a) Explanation: This is a binomial probability problem with \(n = 10\), \(p = 0.5\), and \(k = 5\). \(P(X = 5) = \binom{10}{5}(0.5)^5(0.5)^5 = \binom{10}{5}(0.5)^{10}\) \(\binom{10}{5} = 252\) \(P(X = 5) = 252 \times \frac{1}{1024} \approx 0.2461\) Option (b) incorrectly assumes equal probability for all outcomes. Options (c) and (d) reflect computational errors.
Q6: Which of the following situations can be modeled by a binomial distribution? (a) Drawing 5 cards from a deck without replacement and counting the number of aces (b) Rolling a fair die 20 times and counting the number of sixes (c) Measuring the heights of 30 students (d) Recording the time it takes for 10 students to finish a test
Solution:
Ans: (b) Explanation: A binomial distribution requires independent trials with two outcomes and constant probability. Rolling a die 20 times with independent trials and counting sixes (success) vs. not sixes (failure) fits perfectly. Option (a) violates independence (sampling without replacement). Options (c) and (d) involve continuous measurements, not binary outcomes.
Q7: If \(X \sim B(25, 0.2)\), what is the standard deviation of \(X\)? (a) 2 (b) 4 (c) 5 (d) 3
Solution:
Ans: (a) Explanation: The standard deviation is \(\sigma = \sqrt{np(1-p)}\). \(\sigma = \sqrt{25 \times 0.2 \times 0.8} = \sqrt{4} = 2\) Option (b) is the variance, not standard deviation. Options (c) and (d) result from calculation errors or using the mean.
Q8: For a binomial distribution, if \(n = 50\) and \(p = 0.1\), what is \(P(X \geq 1)\) most easily calculated as? (a) \(1 - P(X = 0)\) (b) \(P(X = 1) + P(X = 2) + ... + P(X = 50)\) (c) \(\binom{50}{1}(0.1)^1(0.9)^{49}\) (d) \(np\)
Solution:
Ans: (a) Explanation: Using the complement rule is the most efficient method: \(P(X \geq 1) = 1 - P(X = 0)\). \(P(X = 0) = \binom{50}{0}(0.1)^0(0.9)^{50} = (0.9)^{50}\) While option (b) is theoretically correct, it requires 50 calculations. Option (c) only gives \(P(X = 1)\), and option (d) gives the mean, not a probability.
Section B: Fill in the Blanks
Q9: The formula for the probability of exactly \(k\) successes in a binomial distribution is \(P(X = k) = \) __________.
Solution:
Ans: \(\binom{n}{k}p^k(1-p)^{n-k}\) Explanation: This is the binomial probability formula, where \(\binom{n}{k}\) is the number of ways to choose \(k\) successes from \(n\) trials, \(p^k\) is the probability of \(k\) successes, and \((1-p)^{n-k}\) is the probability of \(n-k\) failures.
Q10: In a binomial distribution, the mean is calculated using the formula __________ .
Solution:
Ans: \(\mu = np\) Explanation: The expected value or mean of a binomial random variable is the product of the number of trials \(n\) and the probability of success \(p\).
Q11: For a binomial experiment, each trial must be __________ of the other trials.
Solution:
Ans: independent Explanation: One of the four conditions for a binomial distribution is that trials must be independent, meaning the outcome of one trial does not affect the outcomes of other trials.
Q12: The variance of a binomial distribution is given by the formula __________ .
Solution:
Ans: \(\sigma^2 = np(1-p)\) Explanation: The variance measures the spread of the distribution and is calculated by multiplying the number of trials \(n\), the probability of success \(p\), and the probability of failure \((1-p)\).
Q13: If a binomial random variable has \(n = 30\) trials and probability of success \(p = 0.5\), then the mean is __________ .
Solution:
Ans: 15 Explanation: Using the formula \(\mu = np\), we calculate: \(\mu = 30 \times 0.5 = 15\). This represents the expected number of successes in 30 trials.
Q14: In binomial notation, \(X \sim B(n, p)\), the letter \(n\) represents the __________ .
Solution:
Ans: number of trials Explanation: In the binomial distribution notation \(X \sim B(n, p)\), \(n\) represents the fixed number of trials in the experiment, while \(p\) represents the probability of success on each trial.
Section C: Word Problems
Q15: A basketball player has a free throw success rate of 75%. If she attempts 12 free throws during a game, what is the probability that she makes exactly 9 of them? Round your answer to four decimal places.
Solution:
Ans: This is a binomial probability problem with \(n = 12\), \(p = 0.75\), and \(k = 9\). Using the formula: \(P(X = 9) = \binom{12}{9}(0.75)^9(0.25)^3\) \(\binom{12}{9} = \binom{12}{3} = \frac{12 \times 11 \times 10}{3 \times 2 \times 1} = 220\) \((0.75)^9 \approx 0.0751\) \((0.25)^3 = 0.0156\) \(P(X = 9) = 220 \times 0.0751 \times 0.0156 \approx 0.2581\) Final Answer: 0.2581
Q16: A quality control inspector examines 20 randomly selected items from a production line. Historical data shows that 10% of items are defective. What is the expected number of defective items in the sample?
Solution:
Ans: This is a binomial distribution with \(n = 20\) and \(p = 0.10\). The expected value is calculated using: \(\mu = np\) \(\mu = 20 \times 0.10 = 2\) Final Answer: 2 defective items
Q17: A multiple-choice test has 15 questions, each with 4 answer choices. If a student guesses randomly on all questions, what is the probability that the student gets at least 12 questions correct? Round your answer to four decimal places.
Solution:
Ans: This is a binomial distribution with \(n = 15\), \(p = 0.25\), and we need \(P(X \geq 12)\). \(P(X \geq 12) = P(X = 12) + P(X = 13) + P(X = 14) + P(X = 15)\)
Q18: A survey shows that 60% of teenagers use social media daily. If a random sample of 25 teenagers is selected, find the variance of the number of teenagers who use social media daily.
Solution:
Ans: This is a binomial distribution with \(n = 25\) and \(p = 0.60\). The variance is calculated using: \(\sigma^2 = np(1-p)\) \(\sigma^2 = 25 \times 0.60 \times 0.40\) \(\sigma^2 = 25 \times 0.24 = 6\) Final Answer: 6
Q19: A coin is flipped 8 times. Assuming the coin is fair, what is the probability of getting exactly 3 heads? Express your answer as a fraction in simplest form.
Solution:
Ans: This is a binomial distribution with \(n = 8\), \(p = 0.5\), and \(k = 3\). \(P(X = 3) = \binom{8}{3}(0.5)^3(0.5)^5 = \binom{8}{3}(0.5)^8\) \(\binom{8}{3} = \frac{8!}{3!5!} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = 56\) \(P(X = 3) = 56 \times \frac{1}{256} = \frac{56}{256} = \frac{7}{32}\) Final Answer: \(\frac{7}{32}\) or 0.21875
Q20: In a certain city, 30% of residents support a new policy. If 10 residents are randomly selected, what is the standard deviation of the number of residents who support the policy? Round your answer to two decimal places.
Solution:
Ans: This is a binomial distribution with \(n = 10\) and \(p = 0.30\). First, calculate the variance: \(\sigma^2 = np(1-p)\) \(\sigma^2 = 10 \times 0.30 \times 0.70 = 2.1\) The standard deviation is: \(\sigma = \sqrt{2.1} \approx 1.45\) Final Answer: 1.45 residents
The document Worksheet (with Solutions): Binomial Random Variables is a part of the Grade 9 Course Statistics & Probability.
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