1)
There are four elements in this particular probability space: S={a,b,c,d}. Their probabilities are, respectively, .1, .2, .3 and .4. The values of the random variable X, the amount we win or lose for each outcome of the game, are X(a)=1, X(b)=2, X(c)= 3 and X(d)=-4. Compute the mean, variance and standard deviation of X. What is Pr(X>0)? Let Y be the result of playing the game twice, independently. (Y is the sum of X for the first play and X for the second play.) Write down the appropriate probability space for playing the game twice and compute the mean, variance and standard deviation for Y from this. (There are 16 elements in this sample space so you will have to do some arithmetic.) There is a simpler way of computing these three numbers- do it the simpler way and check that you get the same answers. What is Pr(Y>0)? Suppose we play the game 10000 times- let Z be the total we win or lose. Compute the mean, variance and standard deviation of Z. Then Compute Pr(Z>-20000, Pr(Z>0), Pr(Z<-2250). (You may need to use Appendix A for some of this.)
2)
We are interested in words of length 3 using the letters a, b, …, z. (Assume “vowel” means a, e , I, o or u.) How many words are possible? How many words are there where no letter is repeated? How many words are there where a vowel may not be repeated. (For example, bbb is OK but aba is not.) How many words do not have two consecutive vowels? (For instance, aeb is not OK). If we look at all possible 3-letter words, as in the beginning of this question, what is the expected number of vowels? What if we look at three-letter words where a letter may not be repeated- what is the expected number of vowels?
3)
We are playing a game with a “deck” of cards that are numbered 1, 2, …, 100, one of each. A “hand” consists of 5 cards; the order in which the cards are “dealt” does not matter. How many hands are possible? How many hands have only even cards? The “value” of a hand is the sum of the five numbers in it. How many hands have an even value? What is the expected value of a hand? What is the smallest possible value of a hand? How many hands have a value of 19? What is the expected number of even cards in a hand?
4)
Suppose we roll a pair of dice. (Each die is numbered one through six. Assume the six numbers are equally likely and that the dice are rolled independently.) What is the probability that the sum of the two numbers is even? Suppose we keep rolling until we DON’T get a pair and then add the last two numbers. What is the chance that the sum is even in this case? (For instance, if we get (1,1) to start with, we roll again. If we then get (2,2) we roll yet again. If we then get (4,2) we then add the numbers.) If we roll a pair of dice, what is the chance that the total is 7? Suppose we roll one single die. What is the chance that the number we get is 1? What if we have to keep rolling until we get a 1? (For instance, we might get a 1 on the first roll, but the chance of that happening is only 1/6.) What is the chance that it takes exactly five rolls to get a 1? (This means four non-ones followed by a one.) What is the expected number of rolls to get a 1? (This part may be difficult but you should be able to at least make a reasonable guess as to the answer.) If you flip a coin until you get a “head” what is the expected number of flips? (Again, you should be able to at least make a reasonable guess.)
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