Question

Suppose the number of TV's in a household has a binomial distribution with parameters n=20 and p=60%. Find the probability of a household having: (a) 12 or 14 TV's (b) 12 or fewer TV's (c) 18 or more TV's (d) fewer than 14 TV's (e) more than 12 TV's A quiz consists of 20 multiple-choice questions, each with 6 possible answers. For someone who makes random guesses for all of the answers, find the probability of passing if the minimum passing grade is 40%. P( pass )= Suppose it is believed that the probability a patient will recover from a disease following medication is 0.8. In a group of twenty such patients, the number who recover would have mean and variance respectively given by (to one decimal place): Mean: Variance:

Answer 1

Let's break down your questions step by step:

(a) Probability of a household having 12 or 14 TVs:
To find this probability, you can calculate the probability of having exactly 12 TVs and exactly 14 TVs separately, and then add them together. The probability mass function for a binomial distribution is given by:
P(X = k) = (n choose k) * p^k * (1-p)^(n-k)

For k = 12:
P(X = 12) = (20 choose 12) * (0.6)^12 * (0.4)^8

For k = 14:
P(X = 14) = (20 choose 14) * (0.6)^14 * (0.4)^6

Add these probabilities to get the answer.

(b) Probability of a household having 12 or fewer TVs:
You can calculate this by summing up the probabilities of having 0, 1, 2, ..., 12 TVs.

P(X ≤ 12) = Σ [P(X = k) for k = 0 to 12]

(c) Probability of a household having 18 or more TVs:
You can calculate this by summing up the probabilities of having 18, 19, or 20 TVs.

P(X ≥ 18) = Σ [P(X = k) for k = 18 to 20]

(d) Probability of a household having fewer than 14 TVs:
You can calculate this by summing up the probabilities of having 0, 1, 2, ..., 13 TVs.

P(X < 14) = Σ [P(X = k) for k = 0 to 13]

(e) Probability of a household having more than 12 TVs:
You can calculate this by subtracting the probability of having 12 or fewer TVs from 1.

P(X > 12) = 1 - P(X ≤ 12)

For the quiz question:
The probability of passing a multiple-choice quiz with 20 questions and 6 possible answers per question by random guessing can be found using the binomial distribution. Here, n = 20 (number of trials) and p = 1/6 (probability of guessing correctly for each question). You want to find the probability of getting 40% or more correct answers, which means passing (assuming a passing grade is achieved with 40% correct).

P(pass) = P(X ≥ 0.4 * 20) = P(X ≥ 8)

You can calculate this using the binomial distribution as mentioned earlier.

For the patient recovery question:
The mean (expected value) of the number of patients who recover from the disease in a group of 20 can be calculated as n * p = 20 * 0.8 = 16.

The variance can be calculated as n * p * (1-p) = 20 * 0.8 * 0.2 = 3.2.

So, to one decimal place:
Mean: 16.0
Variance: 3.2