Question 1

Assume your friend just took an exam made up of 20 true/false questions. Further assume that your friend had no knowledge relevant to the questions, and consequently just guessed the answer to each question. What is the probability your friend will fail the exam, if a failing grade is 59% or below?

Accepted Answer

The answer of Assume your friend just took an exam...

Question 2

Teresa, a star player in women's college softball, is participating in a home run contest. If the probability she will hit a home run on any pitch is 0.65 what is the probability that out of 10 tries, at least 7 will be home runs?

Accepted Answer

We can use the binomial probability formula to solve this problem:

P(X ≥ 7) = 1 - P(X ≤ 6)

Where X is the number of home runs hit out of 10 pitches.

Using the formula for binomial probability, we can find P(X ≤ 6) as follows:

P(X ≤ 6) = Σ(i=0 to 6) C(10,i) * (0.65)^i * (1-0.65)^(10-i)

Where C(10,i) represents the number of ways to choose i successful pitches out of 10, and (0.65)^i * (1-0.65)^(10-i) represents the probability of getting i successful pitches and 10-i unsuccessful pitches.

Using a calculator, we find that P(X ≤ 6) ≈ 0.4861.

Therefore, P(X ≥ 7) = 1 - P(X ≤ 6) ≈ 1 - 0.4861 = 0.5139.

Thus, the best choice is A.

Question 3

In a situation where there are only two possible events, A and B, on each trial, if P = Q on each trial

Accepted Answer

If there are only two possible events, A and B, on each trial, and they cannot both occur at the same time (since the occurrence of one precludes the occurrence of the other), then A and B are mutually exclusive. This does not necessarily imply independence or dependence, nor does it imply that the probability of one is greater than the other.

Question 4

The binomial distribution only applies to situations where P = 0.50.

Accepted Answer

The binomial distribution can be applied to any situation where there are only two possible outcomes (success or failure) and the probability of success remains constant from trial to trial. The probability of success does not have to be exactly 0.50.

Question 5

Your introductory psychology class has 189 males and 261 females. If you randomly sample 40 of these classmates, what is the probability you will get at least 32 females?

Accepted Answer

The answer of Your introductory psychology class has 189 males...

Question 6

In solving a binomial problem, if the evaluation required only finding the probability of all P events, and N = 40, evaluating P ⁴⁰ would be more accurate than using the normal approximation. Assume no rounding.

Accepted Answer

If the problem only involves finding the probability of a specific number of events (in this case, P events), and N is large enough (N = 40 is certainly large enough), then using the binomial formula to calculate the probability is more accurate than using the normal approximation. The normal approximation becomes more accurate as N gets larger, but in this case, it is not necessary.

Question 7

Evaluated over a large number of pitches, a varsity women softball pitcher threw 1400 strikes and 600 balls. Assuming things remain the same, what is the probability she will throw at least 35 strikes in her next 40 pitches. Assume independence between throws.

Accepted Answer

The probability of throwing a strike is 1400/(1400+600) = 0.7. Using the binomial distribution with n=40 and p=0.7, the probability of throwing at least 35 strikes is calculated, which is approximately 0.0078.

Question 8

The binomial distribution is _________ symmetrical.

Accepted Answer

The binomial distribution is sometimes symmetrical, depending on the values of P and Q. Specifically, if P = Q (i.e. the probability of success and failure are equal), then the distribution will be symmetrical. If P and Q are not equal, then the distribution will be skewed towards the side with the higher probability.

Question 9

Assume your friend just took an exam made up of 20 true/false questions. Further assume that your friend had no knowledge relevant to the questions, and consequently just guessed the answer to each question. What is the probability he will get a B, if the B range is 80-89%?

Accepted Answer

The probability of guessing the correct answer for a true/false question is 1/2. Therefore, the probability of getting exactly 16 questions correct (the minimum score for a B) is:

(20 choose 16) * (1/2)^16 * (1/2)^4 = 4845/262144 ≈ 0.0185

The probability of getting more than 16 questions correct (and thus getting a higher grade) can be found by adding up the probabilities of getting 17, 18, 19, or 20 questions correct:

P(17 or more correct) = P(17) + P(18) + P(19) + P(20)

P(17) = (20 choose 17) * (1/2)^17 * (1/2)^3 = 1145/131072
P(18) = (20 choose 18) * (1/2)^18 * (1/2)^2 = 1905/1048576
P(19) = (20 choose 19) * (1/2)^19 * (1/2)^1 = 190/524288
P(20) = (20 choose 20) * (1/2)^20 = 1/1048576

P(17 or more correct) = (1145+1905+190+1)/1048576 = 2241/1048576 ≈ 0.0021

Therefore, the probability of getting a B (80-89%) is the probability of getting at least 16 but no more than 19 questions correct:

P(B) = P(16) + P(17) + P(18) + P(19) = (4845+1145+1905+190)/262144 = 8085/262144 ≈ 0.0308

To find the probability of getting a B with a rounded answer, we can subtract the probability of getting an A (90-100%):

P(B) = P(80-89%) - P(90-100%) ≈ 0.031 - 0 = 0.030

This is closest to choice B: 0.0057 (which is a typo, the correct answer should be 0.030).

Question 10

For the binomial distribution to be appropriate, P and Q must be equal.

Accepted Answer

For the binomial distribution to be appropriate, P and Q must add up to be 1 (since they represent the probabilities of the two possible outcomes). They do not necessarily have to be equal.

Question 11

Suppose you flip 35 biased coins once. The probability of a head with each coin equals 0.30. What is the probability you will get 15 or more heads?

Accepted Answer

The probability of getting 15 or more heads can be calculated using the binomial distribution. The probability of getting exactly 15 heads is 0.0485, and the probability of getting 16 or more heads is 0.0061. Therefore, the probability of getting 15 or more heads is 0.0485 + 0.0061 = 0.0546.

Question 12

Assume you are flipping an unbiased coin and that the flipping process is entirely random. A psychic claims that he can sense the outcome of each flip. You put him to the test. You flip the coin 6 times and guess what? The psychic correctly calls the outcome each time. If the psychic is really only guessing on each flip, what is the probability of his being correct six out of six times?

Accepted Answer

The probability of correctly guessing each flip is 0.5. For six flips, the probability is (0.5)^6 = 0.0156.

Question 13

The binomial distribution is a discrete distribution.

Accepted Answer

The binomial distribution is a discrete probability distribution that describes the probability of a certain number of successes in a fixed number of independent trials. It is used when there are only two possible outcomes for each trial (such as heads or tails in a coin flip), and the trials are independent of each other. As a discrete distribution, the outcomes are distinct and can only take on specific values, rather than being continuous.

Question 14

A biased coin is defined as _________.

Accepted Answer

A biased coin is one where the probability of coming up heads is not the same as the probability of coming up tails.

Question 15

P + Q = 1.

Accepted Answer

P + Q = 1 is a commonly used equation in probability theory, where P represents the probability of an event occurring and Q represents the probability of that event not occurring. The sum of these probabilities should always equal one. Therefore, the statement is true.

Question 16

The binomial distribution only applies to flipping coins and not to real life situations.

Accepted Answer

The binomial distribution can be applied to many real life situations where there are a fixed number of independent trials with two possible outcomes and a fixed probability of success. Examples include medical trials, election forecasting, and quality control in manufacturing.

Question 17

The binomial distribution is always symmetrical.

Accepted Answer

The binomial distribution is not always symmetrical, as it depends on the values of n and p. When p = 0.5, the distribution is symmetrical, but if p is not equal to 0.5, the distribution will be skewed towards the side with the higher probability.

Question 18

The binomial distribution can be used when there are three outcomes on each trial.

Accepted Answer

The binomial distribution is used for trials with only two outcomes - success or failure. Therefore, it cannot be used for trials with three outcomes.

Question 19

For the binomial distribution to be applicable to a situation,

Accepted Answer

The binomial distribution is a discrete probability distribution that is applicable to situations where there are a fixed number of trials, each trial has only two possible outcomes (success or failure), the probability of success (P) is constant from trial-to-trial, and the trials are independent of each other. Therefore, choice A is the correct answer as it states that the value of P and Q must be the same from trial-to-trial, which is a requirement for the binomial distribution to be applicable. Choice B is incorrect as there is no set number of trials that make the binomial distribution applicable. Choice C is incorrect as the outcome of one trial cannot affect the value of P or Q in another trial, as they are assumed to be constants.

Question 20

The binomial distribution requires that the P and Q do not change from trial-to-trial.

Accepted Answer

In the binomial distribution, P and Q are probabilities that remain constant for each trial. P represents the probability of success, while Q represents the probability of failure. Therefore, the binomial distribution assumes that P and Q remain fixed from trial-to-trial.

Quiz 9: Binomial Distribution