CS70 Discrete Mathematics for Computer Science, Summer 2011Section 101. A roll of the diceConsider a single roll of two dice, one red and one blue.(a) Let R be the value of the red die. What is the distribution of R? What is the expectation of R?(b) Let M be the maximum of the numbers on the two dice. What is the distribution of M ? What is theexpectation of M?(c) How do the distribution and expectation of M compare to that of R?(d) Are the events R = 6 and M = 6 disjoint? Are they independent? What about the events R = 6 andM = 1?[Hint: Recall thatnXi=1i =n(n + 1)2andnXi=1i2=n(n + 1)(2n + 1)6.]2. To pay or not to pay?Reese Prosser never puts money in a 25-cent parking meter in Hanover. He assumes that there is a probabilityof 0.05 that he will be caught. Assume each offense that is caught costs him $10. Under his assumptions:(a) How does the expected cost of parking 10 times without paying the meter compare with the cost ofpaying the meter each time?(b) If he parks at the meter 10 times, what is the probability that he will have to pay more than the totalamount he could end up saving by not putting the money?3. Colorful Marketing LanguageA candy factory has an endless supply of red, orange and yellow jelly beans. The factory packages the jellybeans into jars of 100 jelly beans each, with each possible combination of colors in the jar being equallylikely. (One possible color combination, for example, is a jar of 56 red, 22 orange, and 22 yellow jellybeans.) As a marketing gimmick, the factory claims that the number of possible combinations is so largethat there is negligible probability of finding two jars with the same color combination. You are skepticalabout this claim and decide to do some calculations to test it.(a) Find n, the number of different possible color combinations of jelly beans in a single jar.(b) In terms of n, write down the probability that two jars of jelly beans have different color combinations.(c) Again in terms of n, write down the probability that m jars of jelly beans all have different colorcombinations. [NOTE: You do not need to simplify your expression.](d) Approximately how many jars of jelly beans would you have to buy until the probability of seeing twojars with the same color combination is at least12? [NOTE: You should state your answer only as anorder of magnitude (i.e., 10, 100, 1000, . . .). You may appeal to any result from class in order to deriveyour estimate; it should not be necessary to perform a detailed calculation.]4. Say What?In a binary communication channel the transmitter sends zero or one, but at the receiver there are threepossibilities: a zero is received, a one is received, and an undecided bit is received (which means that thereceiver will ask the transmitter to repeat the bit). Define the event T1= { 1 is sent } and T0= { 0 is sent} and assume that they are equally probable. At the receiver we have three events: R1= { 1 is received },R0= { 0 is received }, Ru= { cannot decide the bit }. We assume that we have the following conditionalprobabilities: Pr[R0|T0] = Pr[R1|T1] = 0.9, Pr[Ru|T0] = Pr[Ru|T1] = 0.09.(a) Find the probability that a transmitted bit is received as undecided.(b) Find the probability that a bit is received in error (error means sending one while receiving zero ORsending zero while receiving one).(c) Given that we received a zero, what is the conditional probability that a zero was sent? What is theconditional probability that a one was sent?5. Locked OutYou just rented a large house and the realtor gave you five keys, one for the front door and the other four foreach of the four side and back doors of the house. Unfortunately, all keys look identical, so to open the frontdoor, you are forced to try them at random.Find the distribution and the expectation of the number of trials you will need to open the front door, underthe following alternative assumptions:(a) after an unsuccessful trial, you mark the corresponding key so that you never try it again, or(b) at each trial, you are equally likely to choose any
View Full Document
Unlocking...