Lesson test: Bayes' theorem

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Bayes' theorem

1.[2p]
Which term is the one people most often leave out when reasoning about evidence informally?

$P (E | H^{c})$ , the chance of the same evidence when the hypothesis is false

Correct

$P (E | H)$ , the chance of the evidence when the hypothesis is true

The answer is: $P(E \mid H^c)$, the chance of the same evidence when the hypothesis is false

$P (H)$ , the prior

The answer is: $P(E \mid H^c)$, the chance of the same evidence when the hypothesis is false

$P (H | E)$ , the posterior

The answer is: $P(E \mid H^c)$, the chance of the same evidence when the hypothesis is false

2.[3p]
One percent of women screened have cancer. The test detects $79.2$ percent of cancers and returns a positive for $9.6$ percent of healthy women. What is the probability that a woman with a positive result has cancer? Give the answer to three decimal places.

CorrectNot quite: 0.077

3.[3p]
Five percent of athletes dope. A test detects $99$ percent of dopers and gives $2$ percent false positives. An athlete tests positive. What is the probability that they doped? Give the answer to three decimal places.

CorrectNot quite: 0.723

4.[2p]
Why does a positive result on an accurate test often leave the probability low?

The false positive rate applies to a much larger group when the condition is rare

Correct

The test is being applied incorrectly

The answer is: The false positive rate applies to a much larger group when the condition is rare

Because sensitivity and specificity cannot both be high

The answer is: The false positive rate applies to a much larger group when the condition is rare

Because the prior is unknown

The answer is: The false positive rate applies to a much larger group when the condition is rare

5.[2p]
A test has sensitivity $0.95$ and a false positive rate $0.10$ . What is the likelihood ratio of a positive result?

CorrectNot quite: 9.5

6.[3p]
Prior odds are $0.05263$ and a positive result has likelihood ratio $12$ . What is the posterior probability? Give the answer to three decimal places.

CorrectNot quite: 0.387

7.[2p]
Updating on evidence $E_{1}$ and then $E_{2}$ gives a different posterior from updating on $E_{2}$ and then $E_{1}$ .

True

The answer is: False

False

Correct

8.[2p]
What happens to a prior of exactly zero under any amount of evidence?

It stays zero, because the odds are multiplied and zero times anything is zero

Correct

It rises slowly towards the likelihood ratio

The answer is: It stays zero, because the odds are multiplied and zero times anything is zero

It becomes undefined

The answer is: It stays zero, because the odds are multiplied and zero times anything is zero

It converges to the true value like any other prior

The answer is: It stays zero, because the odds are multiplied and zero times anything is zero

9.[2p]
Match each quantity in Bayes' theorem to its name.

$P (H)$
$P (E | H)$
$P (H | E)$
$P (E | H) / P (E | H^{c})$

the likelihood ratio
the posterior
the prior
the likelihood

Show the answer

$P (H)$ : the prior $P (E | H)$ : the likelihood $P (H | E)$ : the posterior $P (E | H) / P (E | H^{c})$ : the likelihood ratio

1.[2p] Which term is the one people most often leave out when reasoning about evidence informally?

2.[3p] One percent of women screened have cancer. The test detects 79.2 percent of cancers and returns a positive for 9.6 percent of healthy women. What is the probability that a woman with a positive result has cancer? Give the answer to three decimal places.

3.[3p] Five percent of athletes dope. A test detects 99 percent of dopers and gives 2 percent false positives. An athlete tests positive. What is the probability that they doped? Give the answer to three decimal places.

4.[2p] Why does a positive result on an accurate test often leave the probability low?

5.[2p] A test has sensitivity 0.95 and a false positive rate 0.10. What is the likelihood ratio of a positive result?

6.[3p] Prior odds are 0.05263 and a positive result has likelihood ratio 12. What is the posterior probability? Give the answer to three decimal places.

7.[2p] Updating on evidence E1 and then E2 gives a different posterior from updating on E2 and then E1.

8.[2p] What happens to a prior of exactly zero under any amount of evidence?

9.[2p] Match each quantity in Bayes' theorem to its name.

1.[2p]
Which term is the one people most often leave out when reasoning about evidence informally?

2.[3p]
One percent of women screened have cancer. The test detects $79.2$ percent of cancers and returns a positive for $9.6$ percent of healthy women. What is the probability that a woman with a positive result has cancer? Give the answer to three decimal places.

3.[3p]
Five percent of athletes dope. A test detects $99$ percent of dopers and gives $2$ percent false positives. An athlete tests positive. What is the probability that they doped? Give the answer to three decimal places.

4.[2p]
Why does a positive result on an accurate test often leave the probability low?

5.[2p]
A test has sensitivity $0.95$ and a false positive rate $0.10$ . What is the likelihood ratio of a positive result?

6.[3p]
Prior odds are $0.05263$ and a positive result has likelihood ratio $12$ . What is the posterior probability? Give the answer to three decimal places.

7.[2p]
Updating on evidence $E_{1}$ and then $E_{2}$ gives a different posterior from updating on $E_{2}$ and then $E_{1}$ .

8.[2p]
What happens to a prior of exactly zero under any amount of evidence?

9.[2p]
Match each quantity in Bayes' theorem to its name.