Q: A box contains 2 white balls, 2 blue balls, and 2 red balls. If a ball is drawn at random, what is the probability it is blue?

$$P(\text{blue}) = \frac{2}{6} = \frac{1}{3}$$

Question 1

Which of the following statements best distinguishes a discrete random variable from a continuous random variable?

Accepted Answer

A discrete random variable can only take values that are countable, such as whole numbers, while a continuous random variable can take any value within a given interval.

Question 2

A hospital records the number of patients arriving at the emergency room per hour. Which probability distribution is most appropriate to model this scenario?

Accepted Answer

Poisson distribution

Question 3

Suppose a fair die is rolled. What is the expected value (mean) of the number showing on the die?

Accepted Answer

$$\mu = \frac{1+2+3+4+5+6}{6} = 3.5$$

Question 4

A new drug has a 5% chance of causing side effects. If 50 patients are treated, what is the probability that exactly 2 patients experience side effects? (Use the binomial formula)

Accepted Answer

$$P(X=2) = \binom{50}{2} (0.05)^2$ (0.95$$)^{48}$$

Question 5

Which of the following is NOT an assumption of the binomial distribution?

Accepted Answer

The number of successes must be greater than the number of failures.

Question 6

A nurse checks patient temperatures, which are normally distributed with mean $$\mu = 37$$°C and standard deviation $$\sigma = 0.5$$°C. What is the probability that a randomly selected patient has a temperature above 38°C?

Accepted Answer

Find $$z = \frac{38 - 37}{0.5} = 2$$. $$P(X \u003e 38) = 1 - P(Z \u003c 2) = 1 - 0.9772 = 0.0228$$

Question 7

Which formula gives the variance of a binomial distribution with $$n$$ trials and probability of success $$p$$?

Accepted Answer

$$\$$sigma^2$$ = np(1-p)$$

Question 8

A hospital monitors the length of stay (in days) for post-surgery patients, which follows $$X \sim N(5, 4). $$What percentage of patients stay longer than 7 days?

Accepted Answer

Find $$z = \frac{7-5}{2} = 1$$. $$P(X \u003e 7) = 1 - P(Z \u003c 1) = 1 - 0.8413 = 0.1587$$

Question 9

Which of the following is TRUE about the probability density function (PDF) of a continuous random variable?

Accepted Answer

The total area under the PDF curve is always equal to 1.

Question 10

A blood test result is normally distributed with mean $$\mu = 120 mg/dL $$and $$\sigma = 15 mg/dL. $$What is the z-score for a patient with a result of 145 mg/dL?

Accepted Answer

$$z = \frac{145 - 120}{15} = 1.67$$

Question 11

Which scenario is best modeled by a uniform continuous distribution?

Accepted Answer

The time a nurse randomly checks a patient's blood pressure between 2:00 PM and 3:00 PM.

Question 12

A box contains 2 white balls, 2 blue balls, and 2 red balls. If a ball is drawn at random, what is the probability it is blue?

Accepted Answer

$$P(	ext{blue}) = \frac{2}{6} = \frac{1}{3}$$

Question 13

Which of the following is a key property of the normal distribution?

Accepted Answer

It is symmetric about the mean, and mean = median = mode.

Question 14

A Poisson distribution has a mean $$\lambda = 4. $$What is the probability that exactly 2 events occur? (Use the Poisson formula)

Accepted Answer

$$P(X=2) = \frac{4^2$ e$$^{-4}$$}{2!}$$

Question 15

A nurse administers an IV infusion that takes between 30 and 60 minutes, uniformly distributed. What is the probability that a randomly selected infusion takes between 40 and 50 minutes?

Accepted Answer

$$P(40 \u003c X \u003c 50) = \frac{50-40}{60-30} = \frac{10}{30} = 0.333$$

Question 16

If a random variable $$X$$ follows a standard normal distribution, what is the probability that $$-1 \u003c Z \u003c 1$$?

Accepted Answer

$$P(-1 \u003c Z \u003c 1) = 0.682$$

Question 17

A vaccine is 80% effective. If 10 people are vaccinated, what is the expected number of people who develop immunity?

Accepted Answer

$$E[X] = np = 10 	imes 0.8 = 8$$

Question 18

Which of the following real-world scenarios is best modeled by a Poisson distribution?

Accepted Answer

The number of phone calls received by a hospital switchboard in an hour.

Module 7

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