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The MIT team will receive 2 points for a win, 1 for a tie. and 0 for a loss. Find the PMF of the number of points that the team earns over the weekend.
An urn contains n balls, out of which m are red. We select k of the balls at random. What is the probability that i of the selected balls are red?
A well-shuffled 52-card deck is dealt to 4 players. Find the probability that each of the players gets an ace.
An academic department offers 8 lower level courses: {L1 , L2, .... Ls} and 10 higher level course: {H1. H2.....H10}. How many different curricula are possible?
Find the probability that all the rooks are safe from one another, i.e .. that there is no row or column with more than one rook.
In how many different ways can the cars line up? What is the probability that on a given day, the cars will park in such a way that they alternate?
Ninety students, including Joe and Jane, are to be split into three classes of equal size. What is the probability that Joe and Jane end up in the same class?
We draw two balls randomly and simultaneously. Describe the sample space and calculate the probability that the selected balls are of different color.
We assume that every person has an equal probability of being born on any day during the year. What is the probability that each person has a distinct birthday?
Suppose a red ball was drawn each of n times. What is the probability that if we draw a ball one more time it will be red? Estimate the probability for large m.
A coin that has probability of heads equal to p is tossed successively and independently until a head comes. Compute the expected value of the number of tosses.
What is the probability that at least one of his friends will get the correct card? Hint: Use the inclusion-exclusion formula.
PMF of the minimum of several random variables. On a given day. By how much has your expected score improved as a result of playing on three days?
Assume now that the coin is tossed until we obtain a tail that is immediately preceded by a head. Find the PMF and the expected value of the number of tosses.
Find the PMF, the expected value, and the variance of the number of tosses. What is the probability that the last toss of the first coin is a head?
If this number is n, you receive 2n dollars. What is the expected amount that you will receive? How much would you be willing to pay to play this game?
If the probability of finding a golden ticket is p, find the mean and the variance of the number of candy bars you need to eat to find a ticket.
Find the expected number of questions until you are sure about the location of the prize, under each of the following strategies.
Let a and b be positive integers with a = b, and let X be a random variable that takes as values. Find the expected value and the variance of X.
What would be the temperature range for a normal day if temperature were expressed in degrees Fahrenheit?
Find the PMF of the random variable Y = X mod(3). Find the PMF of the random variable Y = 5 mod(X + 1).
Each natural child has equal probability of being a girl or a boy, independent of the other children. Find the PMF of the number of girls out of the 7 children.
How can we generalize to the case where the probabilities of a left and a right pocket selection are p and 1- p, respectively?
Find the PMF of the number of trials you will need to open the door, under the following alternative assumptions: (1) after an unsuccessful trial.
Find the values of p for which n = 5 is better for the Celtics than n = 3. Find the values for p for which n = 2k + 1 is better for the Celtics than n = 2k - 1.