Q: You have 4 cards with values 3, 5, 6, and 9. If you draw 3 cards without replacement, how many possible outcomes are there?

$$4 \times 3 \times 2 = 24 ($$since order matters)

Q: Given $$P(A) = 0.4$$, $$P(B) = 0.3$$, and $$P(A \text{ and } B) = 0.12$$, are events A and B independent?

Yes, because $$P(A \text{ and } B) = P(A) \times P(B)$$

Q: A company sells boxes with a 20% chance of containing a hat. If you buy 100 boxes, what is the probability of receiving 30 or more hats? (Let $$X be $$the number of hats received.)

$$P(X \geq 30) = 1 - P(X \leq 29)$$, where $$X \sim \text{Binomial}(n=100, p=0.2)$$

Q: If you draw two cards without replacement from the deck described above (no Queens or Kings), what is the probability that neither card is a 6?

$$\frac{40}{44} \times \frac{39}{43}$$

Q: How many distinct 7-digit phone numbers are there that do not contain any 0s and do not repeat any digits?

$$9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3$$

Question 1

Suppose a slot machine pays out with probability $$p = \frac{1}{1000} on $$each play, independently. What is the probability that you get your first payout within the first 100 plays?

Accepted Answer

$$1 - (1 - \frac{1}{1000}$$)^{100}$$

Question 2

Sarasota experiences an average of 2,500 lightning strikes per year. Assuming lightning strikes follow a Poisson distribution, what is the probability of observing between 2000 and 2400 strikes in a year?

Accepted Answer

$$P(2000 \leq X \leq 2400) = \sum_{k=2000}^{2400} \frac{e$$^{-2500}$$ 2500^k}{k!}$$

Question 3

You have 4 cards with values 3, 5, 6, and 9. If you draw 3 cards without replacement, how many possible outcomes are there?

Accepted Answer

$$4 	imes 3 	imes 2 = 24 ($$since order matters)

Question 4

Given $$P(A) = 0.4$$, $$P(B) = 0.3$$, and $$P(A 	ext{ and } B) = 0.12$$, are events A and B independent?

Accepted Answer

Yes, because $$P(A 	ext{ and } B) = P(A) 	imes P(B)$$

Question 5

Given $$P(A) = 0.4$$, $$P(B) = 0.3$$, and $$P(A 	ext{ and } B) = 0.12$$, what is $$P(A 	ext{ or } B)$$?

Accepted Answer

$$0.4 + 0.3 - 0.12 = 0.58$$

Question 6

A company sells boxes with a 20% chance of containing a hat. If you buy 100 boxes, what is the probability of receiving 30 or more hats? (Let $$X be $$the number of hats received.)

Accepted Answer

$$P(X \geq 30) = 1 - P(X \leq 29)$$, where $$X \sim 	ext{Binomial}(n=100, p=0.2)$$

Question 7

For the previous problem, what are the mean and standard deviation of the number of hats received from 100 boxes, each with a 20% chance of containing a hat?

Accepted Answer

Mean $$= 20$$, Standard deviation $$= 4$$

Question 8

A variable is normally distributed with mean $$\mu = 135$$ and standard deviation $$\sigma = 15. $$What percentage of the distribution lies between 100 and 148?

Accepted Answer

$$P(100 \u003c X \u003c 148) = P\left(\frac{100-135}{15} \u003c Z \u003c \frac{148-135}{15}\right)$$

Question 9

Suppose you remove all Queens and Kings from a standard deck of 52 cards. If you draw one card, what is the probability that it is not a 6?

Accepted Answer

$$1 - \frac{4}{44} = \frac{40}{44}$$

Question 10

If you draw two cards without replacement from the deck described above (no Queens or Kings), what is the probability that neither card is a 6?

Accepted Answer

$$\frac{40}{44} 	imes \frac{39}{43}$$

Question 11

How many distinct 7-digit phone numbers are there that do not contain any 0s and do not repeat any digits?

Accepted Answer

$$9 	imes 8 	imes 7 	imes 6 	imes 5 	imes 4 	imes 3$$

Question 12

A restaurant offers 7 appetizers, 10 entrees, and 5 desserts. How many ways are there to order a meal consisting of 1 appetizer, no entrees, and 3 different desserts?

Accepted Answer

$$7 	imes \binom{5}{3}$$

Question 13

A hospital receives an average of 3 emergency calls per hour. Assuming calls are independent and follow a Poisson process, what is the probability that in a given hour, the hospital receives exactly 5 calls?

Accepted Answer

$$P(X=5) = \frac{e$$^{-3}$$ 3^5$}{5!}$$

Question 14

A fair coin is flipped 10 times. What is the probability of getting exactly 7 heads?

Accepted Answer

$$\binom{10}{7} (0.5$$)^{10}$$

Question 15

A bag contains 5 red, 4 blue, and 3 green marbles. If two marbles are drawn without replacement, what is the probability that both are blue?

Accepted Answer

$$\frac{4}{12} 	imes \frac{3}{11}$$

Question 16

A student guesses on a multiple-choice test with 5 questions, each with 4 options. What is the probability the student gets exactly 3 questions correct by guessing?

Accepted Answer

$$\binom{5}{3} (0.25)^3$ (0.75)^2$

Question 17

A normal distribution has mean $$\mu = 100$$ and standard deviation $$\sigma = 20. $$What is the $$z-$$score for $$x = 140$$?

Accepted Answer

$$z = \frac{140 - 100}{20} = 2$$

Question 18

If two events A and B are mutually exclusive and $$P(A) = 0.3$$, $$P(B) = 0.5$$, what is $$P(A 	ext{ or } B)$$?

Accepted Answer

$$0.3 + 0.5 = 0.8$$

STA2023 Practice Exam 2 – Step-by-Step Statistics Guidance

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