How Well Would You Do On A Jury?

1. A woman has been murdered, and her husband is on trial for the crime.

Some years before, the husband had been convicted of domestic violence towards his wife. However, an expert witness states that of all the women who are abused by their boyfriends/husbands, fewer than 1 in 1000 of them end up being murdered by them.

What do these facts tell us about the husband; is he more or less likely to be guilty?

The fact that the husband abused the wife means, other things being equal, that he is more likely to be the murderer

The fact that the husband abused the wife means, other things being equal, that he is less likely to be the murderer

The fact that the husband abused the wife makes no difference one way or the other

The fact that the husband abused the wife may tell us something about whether he is guilty, but the facts as presented are not enough for us to work out what it tells us.

The facts as presented may well be true, but they are not very relevant.

They can tell us something about how likely an abused woman is to be murdered, but that is unhelpful, as in this case we already know that the woman was murdered.

What we need to know about is cases where women are abused by their husbands and then murdered by somebody. In those cases, who is most likely to be the murderer?

In fact, in such cases the husband is the murderer 80 per cent of the time. This information was not presented in the question.

Even this fact may be irrelevant. If more than 80 per cent of murdered women (whether abused or not) are killed by their partners, the presence of abuse cannot tell us anything.

Explanation

The facts as presented may well be true, but they are not very relevant.

They can tell us something about how likely an abused woman is to be murdered, but that is unhelpful, as in this case we already know that the woman was murdered.

What we need to know about is cases where women are abused by their husbands and then murdered by somebody. In those cases, who is most likely to be the murderer?

In fact, in such cases the husband is the murderer 80 per cent of the time. This information was not presented in the question.

Even this fact may be irrelevant. If more than 80 per cent of murdered women (whether abused or not) are killed by their partners, the presence of abuse cannot tell us anything.

2. A man has been murdered. Multiple witnesses saw that the murder was committed by a white man with black hair and a black woman with dyed yellow hair, though none was close enough to see their faces and identify them.

An expert calculates, based on census data, that the odds of finding another interracial couple matching the description are 1 in 12 million.

The suspects are a married couple; the man is white with black hair, and the woman is black with dyed yellow hair. They live in the correct area and have no alibi.

What are the chances that the couple are innocent of the murder?

Less than 1 in 12 million

1 in 12 million

More than 1 in 12 million

We don't have enough information to say

If the murder took place in a reasonably populous area with a strong tradition of racial mixing, there may well be plenty of couples who match the description.

If there is just one other such couple, and we assume that the eyewitness evidence is correct, one of the two couples must be guilty.

Therefore, the probability that the suspects are innocent becomes 1 in 2; that is, 50/50 odds. If there are more such couples, the odds of them being innocent increase further.

Also, the witnesses may have been mistaken. Eyewitness evidence is far less reliable than people think.

Explanation

If the murder took place in a reasonably populous area with a strong tradition of racial mixing, there may well be plenty of couples who match the description.

If there is just one other such couple, and we assume that the eyewitness evidence is correct, one of the two couples must be guilty.

Therefore, the probability that the suspects are innocent becomes 1 in 2; that is, 50/50 odds. If there are more such couples, the odds of them being innocent increase further.

Also, the witnesses may have been mistaken. Eyewitness evidence is far less reliable than people think.

3. Two children have died in the same family. Their parents are on trial, accused of murdering them.

The defence claims that the children both died of Sudden Infant Death Syndrome, or "cot death".

An expert witness testifies that the odds of one child dying of cot death, in a family like the one on trial, is 1 in 8500.

Hence, he argues, the probability that both died of cot death is that probability multiplied by itself: 1 in 73 million.

What is the probability that the two children did both die of cot death?

Less than 1 in 73 million

1 in 73 million

Between 1 in 73 million and 1 in 8500

1 in 8500

More than 1 in 8500

The expert witness is wrong.

His calculation that the odds of a double cot death is 1 in 73 million assumes that the two cot deaths are independent events.

But they may be related; for instance, there may be a genetic effect predisposing the family to cot death.

If one child has died of cot death, the odds that a second one will die may be as high as 1 in 60.

This would mean that the odds of a double death are 1 in 130,000.

Explanation

The expert witness is wrong.

His calculation that the odds of a double cot death is 1 in 73 million assumes that the two cot deaths are independent events.

But they may be related; for instance, there may be a genetic effect predisposing the family to cot death.

If one child has died of cot death, the odds that a second one will die may be as high as 1 in 60.

This would mean that the odds of a double death are 1 in 130,000.

4. A man has been stabbed to death at his home in a city, and a suspect is on trial for the crime.

The murder weapon has been recovered, and a sample of DNA has been obtained from the handle.

The suspect's DNA profile matches that on the handle. An expert witness says that the probability of anyone else being a DNA match is 1 in 3 million.

What are the chances that the DNA on the knife came from someone other than the suspect?

1 in 3 million

More than 1 in 3 million

Less than 1 in 3 million

We don't have enough information to say

1 in 3 million is the likelihood that any innocent person in the population has a DNA profile matching that found on the knife.

If the population is greater than 3 million (as an example, the UK's population is around 60 million), more than 1 person will have that profile.

Hence, the odds that the DNA on the knife did not come from the suspect must be higher than 1 in 3 million.

If you thought the odds were 1 in 3 million, you fell for the Prosecutor's Fallacy.

Explanation

1 in 3 million is the likelihood that any innocent person in the population has a DNA profile matching that found on the knife.

If the population is greater than 3 million (as an example, the UK's population is around 60 million), more than 1 person will have that profile.

Hence, the odds that the DNA on the knife did not come from the suspect must be higher than 1 in 3 million.

If you thought the odds were 1 in 3 million, you fell for the Prosecutor's Fallacy.

5. A man has been murdered, and various pieces of evidence mean that we can be certain that the murderer had a particular disease.

The disease is rare; only 1 in 10,000 people have it.

The suspect has been tested for the disease, using a test that is 99 per cent accurate, and the test was positive.

What is the probability that the suspect really has the disease?

1 in 1: certain

99 in 100

1 in 100

1 in 101

Less than 1 in 101

In a population of 10,000, on average one person will have the disease and they will test positive.

However, because the test is only 99 per cent accurate, 1 per cent of the remaining, healthy population – that is, 100 people – will also test positive.

So if the suspect has tested positive, the odds are 100 in 101 (or slightly more than 99 per cent) that he does not have the disease, and is simply the victim of a false positive.

Turning this around, the odds are 1 in 101 that he has the disease.

Explanation

In a population of 10,000, on average one person will have the disease and they will test positive.

However, because the test is only 99 per cent accurate, 1 per cent of the remaining, healthy population – that is, 100 people – will also test positive.

So if the suspect has tested positive, the odds are 100 in 101 (or slightly more than 99 per cent) that he does not have the disease, and is simply the victim of a false positive.

Turning this around, the odds are 1 in 101 that he has the disease.