Forensic Thinking: A Simple Example

One can only make informed statements about the probability of an event that has been observed or measured objectively. To do otherwise is like trying to guess how likely a die roll will end up "6" without knowing how many sides the die has.

The uncertainty behind such guesswork is obvious. The probability of rolling "6" is 1/6 for a six sided die, 1/20 for a 20 sided die, and zero for a four-sided die. Not knowing anything about the die renders it quite impossible to determine how likely a result would be, the probability would range anywhere from zero to 1/6. The die might even be a trick piece designed to roll "6" more often than any other number, which means the probability is even higher than 1/6.

Now imagine how far off you'd be with more complex problems, like how likely a crime scene DNA profile matches the "real killer" when you weren't present for any of the events that deposited the DNA in the first place.

However all hope is not lost. Let's return to the die example. We need to determine how likely a "6" would be when we roll this unseen die. Let's say that we know that the die in question either has six sides or twenty sides. We can compare the probabilities of rolling a "6" on a six versus a twenty sided die by thinking about the problem like this:

How likely am I to roll a six assuming a six-sided die versus rolling a six assuming a twenty-sided die?

Which would be:

1/6 divided by 1/20 = 20/6 or approximately 3.333

Assuming we have no other evidence regarding six or twenty sided dice being chosen for our experiment, the odds are approximately 3.333 to one in favor of rolling a "6" with a six sided die. Certain results are even more informative than the "6". For example, had we rolled a "7", the probability of getting that result assuming a six-sided die are zero, which would exclude the six sided die from our consideration.

We're not home yet, however. Odds of 3.333 to 1 mean you'll expect to be wrong approximately 23% of the time. What if we rolled the die three times and always came up with numbers of six or less? We'd certainly begin to suspect a six-sided versus a twenty-sided die, But how likely is it?

The probability of rolling any number on a six-sided die is 1/6. However, in this case we're asking the probability of rolling numbers of six or less three times in a row. Since the die can only produce numbers of six or less, the probability for each roll is 1.

1 x 1 x 1

The probability of rolling any number on a twenty-sided die is 1/20. However in this case, we're asking what the probability of rolling any number of six or less on a twenty sided die three times in a row. That becomes

(1/20) + (1/20) + (1/20) + (1/20) + (1/20) + (1/20) = 6/20

(6/20) x (6/20) x (6/20) = 0.027

If we divide the probability of getting numbers of six or less after three rolls on a six-sided die by the probability of the same result on a twenty sided die, we get

1/0.027 = 37.037

Or odds of about 37 to 1. Approximately 97% of the time you'd get this result with a six-sided die versus a twenty-sided die. After these results, we'd be reasonably confident of guessing that the die has six sides and the probability of rolling a "6" is 1/6.