Question 1

Using the probability distribution listed, the mean would be 1.6.

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Question 2

If 1.5% of the bolts made by an automotive factory are defective, what is the probability that in a shipment of 200 bolts, there are 6 defective bolts?

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Question 3

A student takes a 6 question multiple choice quiz with 4 choices for each question. If the student guesses at random on each question, what is the probability that the student gets exactly 4 questions correct?

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Question 4

The formula for the variance of a probability distribution is __________.

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Question 5

A jewelry supplier has a supply of earrings which are 10% platinum. A store owner orders five sets of earrings from the supplier. If the supplier selects the pairs of earrings at random, what is the chance that the jewelry store gets exactly two sets of platinum pairs?

Accepted Answer

We can use the Binomial Distribution formula to solve this problem:

P(X = 2) = (5 choose 2) * 0.1^2 * 0.9^3

where X is the number of sets of platinum pairs that the store gets.

Using a calculator, we can evaluate this expression as:

P(X = 2) = (5!/2!3!) * 0.01 * 0.729 = 0.0729

Therefore, the chance that the store gets exactly two sets of platinum pairs is 0.0729, which corresponds to choice C.

Question 6

A student takes a 10-question, multiple-choice exam with three choices for each question and guesses on each question. Find the probability of guessing exactly 4 out of 10 correctly.

Accepted Answer

The probability of guessing correctly for each question is 1/3, and the probability of guessing incorrectly is 2/3. Since the student is guessing on every question, the probability of guessing exactly 4 out of 10 correctly can be found using the binomial probability formula:

P(X = 4) = (10 choose 4) * (1/3)^4 * (2/3)^6

Using a calculator or a binomial probability table, this simplifies to:

P(X = 4) = 0.228

Therefore, the best choice is D.

Question 7

If a student randomly guesses at 20 multiple-choice questions, find the probability that the student gets exactly four correct. Each question has four possible choices.

Accepted Answer

The probability of getting exactly four correct answers out of 20 multiple-choice questions, each with four possible choices, is 0.190. This can be calculated using the binomial distribution formula:P(X = k) = (n! / (k!(n-k)!)) * p^k * (1-p)^(n-k)where:n is the number of trials (20)k is the number of successes (4)p is the probability of success on each trial (1/4)Plugging these values into the formula, we get:P(X = 4) = (20! / (4!(20-4)!)) * (1/4)^4 * (3/4)^(20-4) = 0.190

Question 8

Find the variance for the values of n and p when the conditions for the binomial distribution are met.
N = 500, p = 0.75

Accepted Answer

The variance of a binomial distribution is calculated using the formula np(1-p). Here, n = 500 and p = 0.75, so the variance is 500 * 0.75 * (1 - 0.75) = 93.75.

Question 9

A coin is tossed five times. Find the probability of getting exactly three heads.

Accepted Answer

The probability of getting one head in a single toss is 1/2 (assuming the coin is fair).
Using the binomial probability formula, we can calculate the probability of getting exactly 3 heads in 5 tosses.
P(3 heads) = (5 C 3) x (1/2)^3 x (1/2)^2 = 10/32 = 0.313
Therefore, the answer is D.

Question 10

A pet supplier has a stock of parakeets of which 20% are blue parakeets. A pet store orders 3 parakeets from this supplier. If the supplier selects the parakeets at random, what is the chance that the pet store gets exactly one blue parakeet?

Accepted Answer

The probability of selecting exactly one blue parakeet can be calculated using the binomial probability formula: P(X=k) = C(n, k) * p^k * (1-p)^(n-k), where n=3, k=1, and p=0.2. This results in P(X=1) = 3 * 0.2^1 * 0.8^2 = 0.384.

Question 11

In a large bag of marbles, 25% of them are red. A child chooses 4 marbles from this bag. If the child chooses the marbles at random, what is the chance that the child gets exactly three red marbles?

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Question 12

Find the mean of the distribution shown below.

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Question 13

If a gambler rolls two dice and gets a sum of 10, he wins $10, and if he gets a sum of three, he wins $20. The cost to play the game is $5. What is the expectation of this game?

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Question 14

A coin is tossed 72 times. Find the standard deviation for the number of heads that will be tossed.

Accepted Answer

The standard deviation for the number of heads can be calculated using the formula:
σ = √(np(1-p))
where n is the number of trials (tosses in this case), p is the probability of getting a head (which is 0.5 for a fair coin).
So, σ = √(72 x 0.5 x 0.5) = 4.24
Therefore, the answer is C.

Question 15

Find the mean for the values of n and p when the conditions for the binomial distribution are met.
N = 700, p = 0.45

Accepted Answer

The mean of a binomial distribution is calculated using the formula: mean = n * p. Here, n = 700 and p = 0.45, so the mean is 700 * 0.45 = 315.

Question 16

A multiple choice quiz consists of 50 questions, each with 4 possible answers. Find the mean for the number of correct answers, if a student guesses on each question.

Accepted Answer

Since there are 4 possible answers for each question, the probability of guessing the correct answer is 1/4. Therefore, the expected value for the number of correct answers on a single question is 1/4. Since there are 50 questions, the expected value for the total number of correct answers is:
Expected value = (1/4) x 50 = 12.5
Therefore, the mean for the number of correct answers, if a student guesses on each question, is 12.5.

Question 17

Compute the probability of X successes.
N = 4, X = 2, p = 0.55

Accepted Answer

Using the binomial distribution formula, the probability of getting exactly 2 successes in 4 trials with a probability of success of 0.55 can be computed as:
P(X=2) = (4 choose 2)(0.55)^2(1-0.55)^2
P(X=2) = (6)(0.3025)(0.2025)
P(X=2) = 0.3678, which is closest to 0.368. Therefore, the best choice is A.

Question 18

Find the standard deviation for the values of n and p when the conditions for the binomial distribution are met.
N = 1000, p = 0.7

Accepted Answer

The standard deviation for a binomial distribution is calculated using the following formula: 
sqrt(n*p*(1-p))

Plugging in the values given, we get: 
sqrt(1000*0.7*(1-0.7)) = 14.5

Therefore, the answer is C.

Question 19

The number of cartoons watched on Saturday mornings by students in Mrs. Kelly's first grade class is shown below.  What is the mean of the data?

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Question 20

The number of cartoons watched on Saturday mornings by students in Mrs. Kelly's first grade class is shown below.  Give the standard deviation for the probability distribution.

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Quiz 5: Discrete Probability Distributions