4.10 to 4.12 Binomial and Geometric Distributions

Multiple Choice

Which of the following best describes a binomial random variable?

• A variable that can take any value within a range

• A variable that counts the number of successes in a fixed number of independent trials

• A variable that measures the time until the first success

• A variable that is always continuous

Multiple Choice

Which of the following is NOT a condition for a binomial setting?

• The trials must be independent.

• The number of trials must be fixed in advance.

• Each trial must have more than two possible outcomes.

• The probability of success must be the same for each trial.

Open Ended

Explain how the binomial random variable is defined and give an example from daily life where it can be used.

Multiple Choice

In the example with 5 children, what is the probability that exactly 2 of them have type O blood?

• 0.2637

• 0.02637

• 0.375

• 0.5

Multiple Choice

The owner of a small convenience store is trying to decide whether to discontinue selling magazines. He suspects that only 5% of the customers buy a magazine and thinks that he might be able to use the display space to sell something more profitable. What is the probability that at least 5 of his first 50 customers buy magazines?

• 0.104
• 0.896
• 0.066
• 0.774

Fill in the Blank

Fill in the blank: The number of ways of arranging k successes among n observations is given by the binomial ___ .

Multiple Select

Select all statements that are true about the binomial probability formula.

• It uses the binomial coefficient to count arrangements of successes.

• It multiplies the probability of k successes and n-k failures.

• It can only be used when the probability of success changes between trials.

• It applies when the number of trials is fixed in advance.

Multiple Choice

Which of the following expressions correctly represents the probability of getting exactly k successes in n binomial trials?

• n!/(k!(n-k)!) * p^k * (1-p)^(n-k)

• n! * p^k * (1-p)^(n-k)

• k!/(n!(n-k)!) * p^k * (1-p)^(n-k)

• n!/(k!(n-k)!) * p^(n-k) * (1-p)^k

Multiple Choice

What is the probability that exactly 3 out of 5 children have type O blood, given each child has a 0.25 probability of having type O blood?

• 0.2637

• 0.3955

• 0.0878

• 0.0147

Multiple Choice

Which of the following are correct formulas for the mean and standard deviation of a binomial random variable X with parameters n and p?

• Mean = np, SD = sqrt(np(1-p))

• Mean = n/p, SD = sqrt(np)

• Mean = np^2, SD = sqrt(n(1-p))

• Mean = n(1-p), SD = sqrt(np(1-p))

Fill in the Blank

The mean number of students who would guess correctly in Mr. Bullard’s class is ___

Open Ended

Why is the probability of finding no defective CDs in a sample of 10 from a shipment of 10,000 not exactly a binomial setting?

Multiple Choice

As n increases in a binomial distribution, what happens to the shape of the distribution?

• It becomes more skewed

• It becomes more uniform

• It becomes more symmetric and bell-shaped

• It becomes bimodal

Multiple Choice

Which of the following conditions must be met to use a Normal approximation for a binomial distribution, as shown in the shopping attitudes example?

• np and n(1-p) must both be at least 10

• The probability of success must be 0.5

• The number of trials must be less than 100

• The distribution must be symmetric

Multiple Choice

A wildlife biologist examines frogs for a genetic trait they suspect may be linked to sensitivity to industrial toxins in the environment. Previous research had established that this trait is usually found in 1 of every 8 frogs. 

2. How many frogs should they expect to collect before they find one with this trait?

• Binomial Distribution

• Geometric Distribution

Multiple Choice

A wildlife biologist examines frogs for a genetic trait they suspect may be linked to sensitivity to industrial toxins in the environment. Previous research had established that this trait is usually found in 1 of every 8 frogs. 

1. The biologist collects and examines a dozen frogs. If the frequency of the trait has not changed, what’s the probability they find the trait in none of the 12 frogs? 

• Binomial Distribution

• Geometric Distribution

Multiple Choice

Suppose a computer chip manufacturer rejects 2% of the chips produced because they fail presale testing.

What’s the probability that the fifth chip you test is the first bad one you find?

• Binomial Distribution

• Geometric Distribution

Multiple Choice

Which of the following random variables is geometric?

• The number of digits I read in a randomly selected row of the random digits table to get a 7

• The number of cards I deal from a well-shuffled deck of 52 cards to get a heart

• The number of 7s in a row of 40 random digits

• The number of 6s I get if I roll a die 10 times

• The number of times I have to roll a single die to get two 6s

Multiple Choice

Seventeen people have been exposed to a particular disease. Each one independently has a 40% chance of contracting the disease. A hospital has the capacity to handle 10 cases of the disease.

What is the probability that the hospital’s capacity will be exceeded?

• 0.989

• 0.011

• 0.965

• 0.035

• 0.092

Multiple Choice

The figure shows the probability distribution of a discrete random variable 𝑋. Which of the following best describes this random variable?

• Binomial with 𝑛 = 8, 𝑝 = 0.8

• Binomial with 𝑛 = 8, 𝑝 = 0.1

• Binomial with 𝑛 = 8, 𝑝 = 0.3

• Geometric with 𝑝 = 0.1

• Geometric with 𝑝 = 0.2

Multiple Choice

Let the random variable X represent the profit made on a randomly selected day by a certain store. Assume that X is approximately Normal with mean $360 and standard deviation $50. The least profitable 10% of days have a profit of at most how many dollars?

• $244

• $296

• $370

• $424

• $476

Multiple Choice

Which of the following random variables is discrete?

• The number of shots made in 10 free-throw attempts.

• The weight of a rhinoceros.

• The species of an insect.

• The length of a needle on a saguaro cactus.

• None of the variables above are discrete.

Multiple Choice

A marketing survey compiled data on the number of personal computers in households. X = the number of computers in a randomly-selected household. If you were to randomly select households, what is the probability it would take exactly 8 selections to find one with 5 computers?

• Approximately 0

• 0.02

• 0.03

• 0.32

• Approximately 1

Multiple Choice

If a data distribution is strongly skewed to the right, what happens to the mean of the data?

• It is pulled to the right of where most of the data lies.

• It is pulled to the left of where most of the data lies.

• The mean stays in the center of where most of the data is.

• We don't care because mean is the mean and the shape of the distribution doesn't affect how we treat the mean...know what I mean.

Multiple Choice

Which statistic measures how spread out the data values are in a data set?

• mean

• mode

• maximum value

• standard deviation

Multiple Choice

The owner of a pet store is trying to decide whether to discontinue selling specialty clothes for pets. She suspects that only 4% of the customers buy those clothes. The owner had 275 customers that day. Assuming this was a typical day, what is the mean and standard deviation of customers who buy specialty clothes? 

• 11 customers, give or take 3.25
• 25 customers, give or take 24.5
• 11 customers, give or take 3.32
• Not enough information.

Multiple Select

Which of the following are required conditions for a geometric setting?

• Binary outcomes

• Independent trials

• Fixed number of trials

• Constant probability of success

Multiple Choice

Which of the following random variables is geometric?

• The number of times I have to roll a die to get two 6s.

• The number of cards I deal from a well-shuffled deck of 52 cards until I get a heart.

• The number of digits I read in a randomly selected row of the random digits table until I find a 7.

• The number of 7s in a row of 40 random digits.

• The number of 6s I get if I roll a die 10 times.

Open Ended

Explain the difference between a binomial setting and a geometric setting. Provide an example of each.

Fill in the Blank

Fill in the blank: In a geometric distribution, the probability that the first success occurs on the k-th trial is given by P(Y = k) = (1-p)^(k-1) ___

Multiple Choice

According to the mean of a geometric distribution, if the probability of success on each trial is 0.2, what is the expected number of trials to get the first success?

• 2

• 5

• 7

• 10

Open Ended

Summarize the key characteristics of a binomial random variable and how its probability distribution is determined.

Multiple Choice

Which of the following conditions must be satisfied to use the normal approximation for a binomial distribution?

• np ≥ 10 and n(1 - p) ≥ 10

• n ≥ 30 and p ≥ 0.5

• np ≤ 5 and n(1 - p) ≤ 5

• n ≥ 100 and p ≤ 0.1

Fill in the Blank

The mean of a geometric random variable Y is ___ .

Open Ended

Explain the difference between a binomial random variable and a geometric random variable, including how their probability distributions are defined.

Open Ended

After learning about binomial and geometric random variables, what questions do you still have or what would you like to explore further about these types of random variables?

Multiple Choice

What is one key difference between binomial and geometric random variables?

• Binomial variables count the number of successes in a fixed number of trials, while geometric variables count the number of trials until the first success.

• Binomial variables are always continuous, while geometric variables are always discrete.

• Geometric variables require a fixed number of trials, while binomial variables do not.

• Binomial variables are used for non-random processes, while geometric variables are used for random processes.