PROBLEM SET 1

1) A probability-minded despot offers a convicted murderer a final chance to gain his release. The prisoner is given twenty chips, ten white and ten black. All twenty are to be placed into two urns, according to any allocation scheme the prisoner wishes, with the one proviso being that each urn contain at least one chip. The executioner will then pick one of the two urns at random and from that urn, one chip at random. If the chip selected is white, the prisoner will be set free; if it is black, he “buys the farm.” Characterize the sample space describing the prisoner’s possible allocation options. (Intuitively, which allocation affords the prisoner the greatest chance of survival?)

2) Let *A*1, *A*2, . . . , *Ak *be any set of events defined on a sample space *S*. What outcomes belong to the event

(*A _{1} *∪

* *

*3) *Let *A *and *B *be any two events. Use Venn diagrams to show that

**(a) **the complement of their intersection is the union of their complements:

(*A*∩*B*)*C *=*AC *∪*BCâ€¨*

**(b) **the complement of their union is the intersection of their complements:

(*A*∪*B*)*C *=*AC *∩*BCâ€¨*

(These two results are known as *DeMorgan’s laws*.)

4) Let *A*,*B*, and *C *be any three events defined on a sample space *S*. Let *N*(*A*), *N*(*B*), *N*(*C*), *N*(*A *∩ *B*), *N*(*A*∩*C*), *N*(*B*∩*C*), and *N*(*A*∩*B*∩*C*) denote the numbers of outcomes in all the different intersections in which *A*, *B*, and *C *are involved. Use a Venn diagram to suggest a formula for *N*(*A *∪ *B *∪ *C*). [*Hint: *Start with the sum *N*(*A*) + *N*(*B*) + *N*(*C*) and use the Venn diagram to identify the “adjustments” that need to be made to that sum before it can equal *N*(*A *∪ *B *∪ *C*).] As a precedent, note that *N*(*A*∪*B*)=*N*(*A*)+*N*(*B*)−*N*(*A*∩*B*). There, in the case of *two *events, subtracting *N*(*A *∩ *B*) is the “adjustment.”

5) For two events *A *and *B *defined on a sample space *S*, *N*(*A*∩*BC*)=15, *N*(*AC *∩*B*)=50, and *N*(*A*∩*B*)=2. Given that *N*(*S*) = 120, how many outcomes belong to neither *A *nor *B*?

6) Urn I contains three red chips and one white chip. Urn II contains two red chips and two white chips. One chip is drawn from each urn and transferred to the other urn. Then a chip is drawn from the first urn. What is the probability that the chip ultimately drawn from urn I is red?

7) An urn contains forty red chips and sixty white chips. Six chips are drawn out and discarded, and a seventh chip is drawn. What is the probability that the seventh chip is red?

8) Recently the U.S. Senate Committee on Labor and Public Welfare investigated the feasibility of setting up a national screening program to detect child abuse. A team of consultants estimated the following probabilities: (1) one child in ninety is abused, (2) a screening program can detect an abused child 90% of the time, and (3) a screening program would incorrectly label 3% of all non abused children as abused. What is the probability that a child is actually abused given that the screening program makes that diagnosis? How does the probability change if the incidence of abuse is one in one thousand? Or one in fifty?

9) An “eyes-only” diplomatic message is to be trans- mitted as a binary code of 0’s and 1’s. Past experience with the equipment being used suggests that if a 0 is sent, it will be (correctly) received as a 0 90% of the time (and mistakenly decoded as a 1 10% of the time). If a 1 is sent, it will be received as a 1 95% of the time (and as a 0 5% of the time). The text being sent is thought to be 70% 1’s and 30% 0’s. Suppose the next signal sent is received as a 1. What is the probability that it was sent as a 0?

Subject | Mathematics |

Due By (Pacific Time) | 05/21/2014 02:59 pm |

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