please do number 55 voll a die twice and record the largest number (if the two r
ID: 3072621 • Letter: P
Question
please do number 55
voll a die twice and record the largest number (if the two rolls give the same the largest number). (a) Given that the first roll gives 1, what is 50. Iconditional probability that the largest number is 3? (b) Given that the first outcome it crives 3, what is the conditional probability that the largest number is 3 I gives 3, what i u two fair dice. Let Ai be the event that the first die gives k, and let B, be the es of n and k are As and B, independent? event that the sum is n. For which val distribution of blood types in the United States according to the "ABO ion" is O:45%, A : 40%, B 11%, and AB 4% Blood is also classified ng to Rh type, which can be negative or positive and is independent of type (the corresponding genes are located on different chromosomes). pulation, about 84% are Rh positive. Sample two individuals at The classifica ac the ABO the U.S. pop m and find the probability that (a) both are A negative, (b) one of them is no o and Rh positive, while the other is not, (e) at least one of them is O positive. id) one is Rh positive and the other is not AB, (e) they have the same ABO type, and () they have the same ABO type and different Rh types ss. Let A, B, and C be independent events. Show that A is independent of both BnC and BUC S4. You are offered to play the following game: A roulette wheel is spun eight times If any of the 38 numbers (0,00,1-36) is repeated, you lose $10, otherwise you win $10. Should you accept to play this game? Argue by computing the relevant probability. 5. Consider the following simplified version of the birthday problem in Example 1.19. Divide the year into "winter half" and "summer half." Suppose that the probability is p that an individual is born in the winter half. What is the proba- bility that two people are born in the same half of the year? For which value of p is this minimized? 56. Consider the birthday problem with two people and suppose that the probability distribution of birthdays is p,. .. , P365. (a) Express the probability that they have the same birthday as a function of the pk. (b) Show that the probability in (a) is minimized for pk365,k 1,2,... 365. If you are familiar with Lagrange I0True storyExplanation / Answer
55)P(that both people born in same half)=P(A) =P(both are born in winter half)+P(both are born in summer half)=p*p+(1-p)*(1-p) =p2+1+p2-2p =2p2-2p+1
for minimization: differentiating above with respect to p:
(d/dp)*P(A) =(d/dp)*(2p2-2p+1) =4p-2
putting (d/dp)*P(A) =0
4p-2 =0
p=2/4 =1/2
hence for miniizing the probability that both are born in same half ; p should be 1/2.
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