Question

In a murder trial in Los Angeles, the prosecution claims that the defendant was cut on the left middle finger at the murder scene, but the defendant claims the cut occurred in Chicago, the day after the murders had been committed. Because the defendant is a sports celebrity, many people noticed him before he reached Chicago. Twenty-two people saw him casually, one person on the plane to Chicago carefully studied his hands looking for a championship ring, and another person stood with him as he signed autographs and drove him from the airport to the hotel. None of these 24 people saw a cut on the defendant’s finger. If in fact he was not cut at all, it would be extremely unlikely that he left blood at the murder scene.

(a) Because a person casually meeting the defendant would not be looking for a cut, assume that the probability is 0.7 that such a person would not have seen the cut, even if it was there. Furthermore, assume that the person who carefully looked at the defendant’s hands had a 0.4 probability of not seeing the cut even if it was there and that the person who drove the defendant from the airport to the hotel had a 0.6 probability of not seeing the cut even if it was there. Given these assumptions, and also assuming that all 24 people looked at the defendant independently of each other, what is the probability that none of the 24 people would have seen the cut, even if it was there? (Do not round intermediate calculations and round your final answer to 4 decimal places.)

Probability          

(b) What is the probability that at least one of the 24 people would have seen the cut if it was there? (Round your answer to 4 decimal places.)

Probability         

(c) Given the result of part b and given the fact that none of the 24 people saw a cut, do you think the defendant had a cut on his hand before he reached Chicago?

Yes or No

Answer 1

Solution

A1 be the event that a person casually meeting the defendant would not have seen the cut

A2 be the event that a person who carefully looked at the defendant’s hands would not have seen the cut

A3 be the event that a person who drove the defendant from the airport to the hotel would not have seen the cut

B be the event the defendant had a cut on the finger.

P(A1/B) = 0.7

P(A2/B) = 0.4

P(A3/B) = 0.6

Back-up Theory

If A and B are independent, P(A ∩ B) = P(A) x P(B) ..…………………………(1)

If A1, A2, ........., Ak are independent events, then

P(A1∩ A2∩, ........., ∩Ak) = P(A1) x P(A2) x ………. x P(Ak) …………………(1a)

Part (a)

Out of 24 persons, one had carefully looked at the defendant’s hands, one drove the defendant from the airport to the hotel and the remaining 22 casually met the defendant.

Also given that all 24 people looked at the defendant independently of each other, vide (1a),

probability none would have seen the cut even it was there

= { P(A1/B)}²² x P(A2/B) x P(A3/B)

= 0.7²² x 0.4 x 0.6

= 9.38E-05 = 0.0000938 = 0.0001 Answer 1

Part (b)

The probability that at least one of the 24 people would have seen the cut if it was there

= 1 - Probability none would have seen the cut even it was there

= 0.9999062 [vide Answer 1]

= 0.9999 Answer 2

Part (c)

Part (b) answer puts the probability of at least one person seeing the cut if it was there very high, virtually equal to 1. So, if none of the 24 people had seen the cut, it gives strong evidence to conclude that there was no cut.

Hence No is the Answer 3

DONE