Question

During the case People v. Collins, the prosecution assumed the following chances:

P(yellow car) = 1/10	P(woman and blond hair) = 1/3
P(man and mustache) = ¼	P(black man and beard) = 1/10
P(woman and ponytail) = 1/10	P(interracial couple in car) = 1/1000

Suppose you are called as a witness for the defense. The following prompts are concerned with probabilities related to the prosecution’s probability claims. Using modern resources to estimate probabilities for events in the relevant sample space (time and place), we find:

P(white) = 0.41	P(black) = 0.1
P(beard) = 0.05	P(mustache) = 0.08
P(female) = 0.5

1) Notice that P(man and mustache) = ¼ given by the prosecution is much larger than the P(mustache) = 0.08 given here. Assume ‘man with mustache’ means ‘man and mustache’. Given that the events ‘man with mustache’ and ‘mustache’ are slightly different and assuming they describe the same population, could both probabilities be correct? Answer ‘yes’ or ‘no’ and explain.

2) Does the larger probability [P(man with mustache) = ¼ vs. P(mustache) = 0.08] given by the prosecution strengthen or weaken their case? Explain.

Answer 1

1)

When we have two events A and B then :

$\dpi{100} P(A\ and\ B ) \leq P(A)$

or

$\dpi{100} P(A\ and\ B ) \leq P(B)$

But

$\dpi{100} P(A\ and\ B ) \ngtr P(A)$

or

$\dpi{100} P(A\ and\ B ) \ngtr P(B)$

In this case,

P(man and mustache) = 1/4 =0.25

P(mustache) = 0.08

Intersection between two events cannot be greater than either of the events.

Since, P(man and mustache) > P(mustache) , which is not possible. Hence, Both probabilities cannot be correct.

2)

Larger probability [P(man with mustache) = 1/4 vs. P(mustache) = 0.08] given by the prosecution will definitely weaken their case because the chances assumed by the prosecution are not possible as Probability for Mustache with Man cannot be greater than probability of person with mustache.

Probability for Mustache with Man can be less than or equal to probability of person with mustache but not greater than probability of person with mustache.