Q: Which of the following best describes the complement rule in probability, as applied in the birthday problem?

The probability of at least one match is $$1 - P(\text{no matches}).$$

Q: Which of the following is the correct formula for the probability of a positive test result in a population?

$$P(POS) = P(S) \times P(POS|S) + P(S^c) \times P(POS|S^c)$$

Question 1

In the classic birthday problem, what is the probability that at least two students in a class of 25 share the same birthday? (Assume 365 days in a year and ignore leap years.)

Accepted Answer

$$0.57$$

Question 2

Which of the following best describes the complement rule in probability, as applied in the birthday problem?

Accepted Answer

The probability of at least one match is $$1 - P(	ext{no matches}).$$

Question 3

A probability model is best described as:

Accepted Answer

A specification of possible outcomes and assumptions for probability calculations.

Question 4

In the space shuttle example, the probability of at least one failure in 100 missions, given a success probability of $$0.971$$ per mission, is calculated as:

Accepted Answer

$$1 - (0.971$$)^{100}$$

Question 5

Which assumption is NOT required for the calculation of the probability of at least one space shuttle failure in 100 missions?

Accepted Answer

The probability of failure changes after each mission.

Question 6

A diagnostic test for a rare disease has sensitivity $$0.96$$ and specificity $$0.93. If $$the disease prevalence is $$0.007$$, what is the probability that a randomly selected person will test positive?

Accepted Answer

$$0.0762$$

Question 7

Given the following: $$P(S) = 0.007$$, $$P(POS|S) = 0.96$$, $$P(NEG|S^c) = 0.93. $$What is the probability that a person who does NOT use drugs tests positive?

Accepted Answer

$$0.07$$

Question 8

Which of the following is the correct formula for the probability of a positive test result in a population?

Accepted Answer

$$P(POS) = P(S) 	imes P(POS|S) + P(S^c) 	imes P(POS|S^c)$$

Question 9

Why might the calculated probability of at least one space shuttle failure in 100 missions be inaccurate in reality?

Accepted Answer

Because the calculation assumes independence and constant probability, which may not hold due to factors like crew experience and equipment quality.

Question 10

Which of the following best describes the purpose of a simulation in probability estimation?

Accepted Answer

To approximate probabilities when analytical calculation is difficult.

Question 11

If a diagnostic test has a high sensitivity but low specificity, what is the likely outcome in a population with low disease prevalence?

Accepted Answer

Many false positives will occur.

Question 12

Which of the following is a correct step in carrying out a simulation to estimate a probability?

Accepted Answer

Describe how to simulate observations of the random phenomenon.

Question 13

In the context of probability models, which statement is TRUE?

Accepted Answer

Probability models are simplifications and may not perfectly reflect reality.

Question 14

A class of 25 students is asked about their birthdays. What is the probability that no two students share a birthday?

Accepted Answer

$$0.43$$

Question 15

Which of the following is a plausible reason why coincidences (such as two people sharing a birthday) are not as unusual as they seem?

Accepted Answer

Because there are a large number of possible pairs and events that can occur.

Question 16

If the sensitivity of a test is $$0.96$$ and the specificity is $$0.93$$, what is the false positive rate?

Accepted Answer

$$0.07$$

Question 17

Which of the following best describes 'sensitivity' in the context of diagnostic testing?

Accepted Answer

The probability that the test is positive given the condition is present.

Question 18

A simulation is run 1000 times to estimate the probability of a rare event. Which of the following is the best way to summarize the results?

Accepted Answer

Calculate the proportion of times the event occurred in the simulation.

Question 19

Which of the following is NOT a step in carrying out a simulation to estimate a probability?

Accepted Answer

Assume the answer without running the simulation.

(Lecture 12) Applying Probability Rules: Models, Examples, and Diagnostic Testing

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