The binomial distribution：練習問題

A student takes a true/false quiz consisting of 10 questions. If the student guesses the answer to each question randomly, what is the probability that they get exactly 7 questions correct?

Let \( X \) be a binomial random variable. If the mean of \( X \) is 4 and the variance of \( X \) is 2.4, find the values of the parameters \( n \) and \( p \).

A random variable \(X\) follows a binomial distribution \(B(n, p)\). Given that the mean of \(X\) is \(12\) and \(P(X=0) = (0.7)^{40}\), find the variance of \(X\).

In a binomial distribution \( B(n, p) \), if \( n=8 \) and \( p=0.4 \), find the probability \( P(X \leq 1) \) correct to 4 decimal places.

A multiple-choice test consists of \(5\) questions. Each question has \(4\) options with only one correct answer. If a student guesses the answers randomly, find the probability that the student gets at least \(4\) questions correct.

A student answers a multiple-choice quiz consisting of 10 questions. Each question has 4 possible answers, only one of which is correct. If the student answers all questions by guessing randomly, what is the expected number of questions the student will answer correctly?

Let \(X\) be a discrete random variable following a binomial distribution \(B(n, p)\). Given that the mean and the variance of \(X\) are 4.8 and 2.88 respectively, find the probability that \(X \le 2\) given that \(X \ge 1\). (Express your answer correct to 4 decimal places.)

Let \(X\) be a discrete random variable following a binomial distribution \(B(n, p)\) with mean 4 and variance 3. Another independent discrete random variable \(Y\) follows a binomial distribution \(B(m, 0.4)\).
(a) Find the values of \(n\) and \(p\).
(b) Find the smallest integer \(m\) such that \(P(Y \ge 1) > 0.95\).
(c) Using the value of \(m\) found in (b), find \(P(X + Y = 1)\). (Express your answer correct to 4 decimal places.)

A small-scale survey is conducted in a local community. It is known that \(30\%\) of the households own a pet. A random sample of \(10\) households is selected.

(a) Find the probability that exactly \(4\) households in the sample own a pet. (2 points)

(b) Find the probability that more than \(2\) households in the sample own a pet. (2 points)

(c) Determine the expected number and the variance of the households that own a pet in this sample. (1 point)

A manufacturing plant produces high-precision sensors. It is known that 8% of the sensors are defective. Sensors are randomly selected and packed into boxes, where each box contains 15 sensors.

(a) Find the probability that a box contains exactly 2 defective sensors.

(b) A box is classified as "Grade A" if it contains at most 1 defective sensor. Find the probability that a box is "Grade A".

(c) A technician randomly selects 10 boxes for inspection. A shipment consisting of these 10 boxes is considered "Premium" if at least 8 of the boxes are "Grade A".
(i) Find the probability that the shipment is "Premium".
(ii) Given that the shipment is "Premium", find the probability that there are exactly 9 "Grade A" boxes in the shipment.

(Give your answers correct to 4 decimal places.)