Q: If the probability of an event is $$p = 0.4$$, what is the probability that the first occurrence of the event happens on the 4th trial? (Geometric distribution)

$$p(x=4) = 0.4 \times (0.6)^3$ = 0.0864$$

Question 1

Suppose you have a box of 100 gumballs: 40 red, 20 blue, 30 green, and 10 yellow. If you draw one gumball at random, what is the probability that it is either red or yellow?

Accepted Answer

$$p = \frac{40+10}{100} = \frac{50}{100} = 0.5$$

Question 2

If you flip a fair coin three times, what is the probability of getting exactly two heads?

Accepted Answer

$$p = \frac{3}{8}$$

Question 3

A survey finds that 40% of Canadians are concerned about job security. If you interview 15 people, what is the probability that exactly 10 are concerned? (Use the binomial formula.)

Accepted Answer

$$p = \binom{15}{10} (0.4$$)^{10}$$ (0.6)^5$

Question 4

If the probability of an event is $$p = 0.2$$, what is the probability that the event does NOT occur in a single trial?

Accepted Answer

$$q = 1 - p = 0.8$$

Question 5

You roll a die and flip a coin. How many possible outcomes are there in the sample space?

Accepted Answer

$$N(S) = 2 	imes 6 = 12$$

Question 6

If the probability of a COVID death for people aged 70 or over is $$p = 0.00568$$, what is the probability that a randomly selected person in this age group does NOT die from COVID?

Accepted Answer

$$q = 1 - 0.00568 = 0.99432$$

Question 7

A Poisson process has an average rate of $$\lambda = 1.5$$ alarms per week. What is the probability of observing exactly 3 alarms in a week?

Accepted Answer

$$p = \frac{e$$^{-1.5}$$ (1.5)^3$}{3!}$$

Question 8

If you want to be 95% confident that you will hear about any issue held by at least 10% of the population, how many people should you interview so that the probability of NOT hearing about the issue is less than 5%?

Accepted Answer

$$n = \lceil \frac{\ln(0.05)}{\ln(0.9)} \rceil$$

Question 9

Given a normal distribution with mean $$\mu = 28.3$$ and standard deviation $$\sigma = 9.5$$, what is the z-score for a raw score of 43?

Accepted Answer

$$z = \frac{43 - 28.3}{9.5} = 1.55$$

Question 10

Which of the following is NOT a condition for using the binomial probability distribution?

Accepted Answer

The probability of success changes from trial to trial.

Question 11

If you draw a yellow gumball from a jar and do NOT replace it, what is the probability that the next gumball you draw is also yellow? (Assume there were originally 10 yellow out of 100 gumballs.)

Accepted Answer

$$p = \frac{9}{99}$$

Question 12

What is the formula for the probability of getting exactly $$r$$ successes in $$n$$ independent Bernoulli trials, each with probability $$p of $$success?

Accepted Answer

$$P(r) = \binom{n}{r} p^r (1-p$$)^{n-r}$$

Question 13

If the probability of an event is $$p = 0.4$$, what is the probability that the first occurrence of the event happens on the 4th trial? (Geometric distribution)

Accepted Answer

$$p(x=4) = 0.4 	imes (0.6)^3$ = 0.0864$$

Question 14

Which of the following best describes the sample space for flipping a coin three times?

Accepted Answer

There are $$2^3 = 8$ possible outcomes.

Question 15

If a student scores 9 on a test with mean $$\mu = 28.3$$ and standard deviation $$\sigma = 9.5$$, what is their z-score?

Accepted Answer

$$z = \frac{9 - 28.3}{9.5} = -2.03$$

Question 16

Which of the following is a correct interpretation of a z-score of 1.0 in a normal distribution?

Accepted Answer

The value is one standard deviation above the mean.

Probability and Hypothesis Testing: Foundations and Applications in Statistics

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