NAME_______________________

1. A class in advanced physics is comprised of 8 juniors, 22 seniors and 10 graduate students. The final grades showed that 3 of the juniors, 10 of the seniors and 5 of the graduate students recieved an A for the course.

a. Draw the Venn Diagram for this situation.


Juniors
Seniors
Graduate

A
3
10
5
18

AC
5
12
5
22


8
22
10
40

b. If a student is chosen at random from this class and is found to have an A, what is the probabiltity that he or she is a junior?

18 total A's, and 3 Juniors with a's, therefore P(J&A) = 3/18 = 1/6th

c. If a junior is chosen at random from this class, what is the probability of that junior not receiving an A?

8 Juniors, so 5 did not get an a, therefore 5/8 probability or .625

AC BC

2. Match the following statements with the appropriate of the Venn Diagram(s), (One, Two, Three), below:

A is independent of BC ___Two_________

AC is positively correlated with B __One_________

BC is negatively correlated with AC ____One________

Prob(A|B) = Prob(A|BC) ____Two________

Prob(A|B) > Prob(A|BC) ____Three________

One
A
AC


Two
A
AC


Three
A
AC

B
5
15
20

B
2
5
7

B
6
1
7

BC
5
5
10

BC
4
10
14

BC
2
5
7


10
20



6
15



8
6

3. Lets Play Monty Hall. Suppose there are 3 doors and the probability that the prize is behind Door 1 is 0.5, Door 2 is 0.35, and Door 3 is 0.15. If you initially pick Door 2,

a) what is the probability of winning with Door 2 if he shows you Door 1?, if he shows you Door 3?


Door 1
Door 2
Door 3

Show 1
0
.175
.15
.325

Show 3
.5
.175
0
.675


.5
.35
.15
1.00

If shows 1 then .175/.325 = .538

If shows 3 then .175/.675 = .259





b) would you be better off switching if he shows you Door 1?, should you switch if he shows you Door 3?

if shows 1 then .15/.325 = .462, therefore do not switch

if shows 3 then .5/.675 = .741 therefore switch

4. Two of three prisoners have been chosen to be executed, but the names of the two are to be kept secret. Prisoner A can't stand the suspense. He says to the guard, "I know you can't tell me if I"m to be executed, but since we both know that one of B and C prisoners will be, surely you can tell me the name of the one of them that will be. It can't hurt, because it really isn't new information." The guard pauses for a moment and then says, "Well, OK, Prisoner C will be executed." Prisoner A goes to sleep happy because his probability of being executed have been reduced from 2/3 to 1/2. True or False? Please explain.

False, because his probability of not being executed, 1/3rd, remains the same whether he knows the new information or not.

5. Using the SIMPLE Search data and graph answer the following:

a) If you could only look in three of the four areas on the first seach, which area would it be best to avoid? Why?

Area 1, since the P(A&D) is the lowest, .005.

b) What is the significance of the intersection of A1 and A3 on seach number 8?

Previously, A3 was the area the most likely had the object in it, after this unsuccessful search it is now A1. P(A1) > P(A2)


c) After many unsuccessful searches, A1 approaches 1.0 on the graph. What does mean and Why does this occur?

P(A1) has the lowest P(D), therefore it after many unsucessful searches it is the most likely location of the object.

d) What is the probability that the object is in Area 1 and detected, P(A1 Ç D), after one simple search is completed?

.0783 - .0704 = .0079

e) What is the probability of the object being in Area 3, P(A3), after two searches are completed?

0.5009 / .6096 = .822

6. Using the SIMPLE Search data and graph answer the following:

a) Which Areas are the best to seach in on search:

Four? __A4______ Five? __A1______ Six? __A1_______

b) Once Area 4 is searched, why doe it take so long, almost 30 searches, until we look there again?

The P(D) is very high in this area, so to look once you should find it when compared to the other areas with lower P(D). Those other areas require more searches since hard to find the object.

c) Calculate the probability that the object is in each area, P(A1), P(A2), P(A3), and P(A4), after one unsuccessful search?

A1: .35/.85 = .412

A2: .30/.85 = .353

A3: .15/.85 = .176

A4: .05/.85 = .059