Q: A medical test for cancer has a true positive rate of 80% and a false positive rate of 10%. If the prevalence of cancer is 1%, what is the probability that a person who tests positive actually has cancer?

$$\frac{0.01 \times 0.8}{0.01 \times 0.8 + 0.99 \times 0.1}$$

Q: A test for a rare disease has a 5% false positive rate and a 90% true positive rate. If the disease prevalence is 2%, what is the probability that a person who tests positive actually has the disease? (Use Bayes’ theorem.)

$$\frac{0.02 \times 0.9}{0.02 \times 0.9 + 0.98 \times 0.05}$$

Question 1

A factory produces items with a defect rate of 15%. If a customer buys 12 items, what is the probability that at least one item is defective? Use the complement rule.

Accepted Answer

$$1 - 0.85$$^{12}$$

Question 2

Which of the following best describes the complement of the event 'at least one'?

Accepted Answer

Getting no occurrences of the event

Question 3

Given $$P(A) = 0.3$$ and $$P(B) = 0.5$$, and $$P(A 	ext{ and } B) = 0.15$$, what is $$P(B|A)$$?

Accepted Answer

$$\frac{0.15}{0.3} = 0.5$$

Question 4

In a pre-employment drug screening, 45 out of 50 drug users tested positive. What is $$P(	ext{positive test} | 	ext{uses drugs})$$?

Accepted Answer

$$\frac{45}{50}$$

Question 5

Why is $$P(B|A) 
eq P(A|B) in $$general?

Accepted Answer

Because the probability of A given B is not the same as the probability of B given A unless in special cases

Question 6

A medical test for cancer has a true positive rate of 80% and a false positive rate of 10%. If the prevalence of cancer is 1%, what is the probability that a person who tests positive actually has cancer?

Accepted Answer

$$\frac{0.01 	imes 0.8}{0.01 	imes 0.8 + 0.99 	imes 0.1}$$

Question 7

What is a posterior probability?

Accepted Answer

A probability value revised using additional information that is later obtained

Question 8

How many different 5-character passwords can be formed using 92 possible characters for each position?

Accepted Answer

$$92^5$$

Question 9

If you randomly select one password from all possible 5-character passwords (using 92 characters), what is the probability of guessing the correct password on the first try?

Accepted Answer

$$\frac{1}{92^5$}$$

Question 10

How many different ways can the letters in the word 'steam' be arranged?

Accepted Answer

$$5!$$

Question 11

In a horse race with 20 horses, how many different trifecta bets (selecting 1st, 2nd, and 3rd place in order) are possible?

Accepted Answer

$$20 	imes 19 	imes 18$$

Question 12

A survey has 10 questions, 2 of which are identical and 3 others are also identical. How many different arrangements of the questions are possible?

Accepted Answer

$$\frac{10!}{2!3!}$$

Question 13

Which formula is used to calculate the number of combinations when selecting $$r$$ items from $$n$$ without replacement and order does not matter?

Accepted Answer

$$\frac{n!}{r!(n-r)!}$$

Question 14

In Florida's Cash 5 lottery, you must select 5 numbers from 1 to 35. How many different tickets are possible?

Accepted Answer

$$\frac{35!}{5!(35-5)!}$$

Question 15

If you buy one ticket in the Cash 5 lottery, what is the probability of winning the jackpot?

Accepted Answer

$$\frac{1}{\frac{35!}{5!(35-5)!}}$$

Question 16

Which of the following is true about permutations?

Accepted Answer

Order matters

Question 17

Which mnemonic helps you remember that order matters in permutations?

Accepted Answer

"Permutations Position"

Question 18

A test for a rare disease has a 5% false positive rate and a 90% true positive rate. If the disease prevalence is 2%, what is the probability that a person who tests positive actually has the disease? (Use Bayes’ theorem.)

Accepted Answer

$$\frac{0.02 	imes 0.9}{0.02 	imes 0.9 + 0.98 	imes 0.05}$$

Question 19

Which of the following is an example of a prior probability?

Accepted Answer

The probability of having cancer before any test results are known

Probability: Complements, Conditional Probability, and Bayes’ Theorem

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