A math class has 25 students. Assuming that all of the students were born in the first half of the year – between January 1st and June 30th – what is the probability that at least two students have the same birthday? Assume that nobody was born on leap day, so there are 181 possible birthdays.

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6. AND Since there are two zeros on the board, let's determine the probability that you would be able to cover them on 2 consecutive turns. This means you'd have to roll a zero and then roll another zero. You can think of this as an AND probability. You need to roll 0 AND then you need to roll 0 again. Since the result of the first roll will have no influence on the result of the second roll, you can use the following rule: P(A and B) = P(A) * P(B) a) Determine P(0 and 0). b) Determine 1 - P(0 and 0). c) Write a sentence describing the meaning of 1 - P( Oand 0 ).

Hat-check problem. Use indicator random variables to solve the following problem, which is known as the hat-check problem. Each of n customers gives a hat to a hat-check person at a restaurant. The hat-check person gives the hats back to the customers in a random order. What is the expected number of customers who get back their own hat?

World Series odds Consider two teams, A and B, playing a series of games until one of the teams wins n games. Assume that the probability ofA winning a game is the same for each game and equal to p, and the probability of A losing a game is q = 1− p. (Hence, there are no ties.) Let P(i, j) be the probability of A winning the series if A needs i more games to win the series and B needs j more games to win the series. a. Set up a recurrence relation for P(i, j) that can be used by a dynamic programming algorithm. b. Find the probability of team A winning a seven-game series if the proba-bility of it winning a game is 0.4. c. Write pseudocode of the dynamic programming algorithm for solving this problem and determine its time and space efficiencies.