Short homework this week because of our exam!
People are arriving at a party one at a time. While waiting for more people to arrive they entertain themselves by comparing their birthdays. Let X be the number of people needed to obtain a birthday match, i.e., before person X arrives no two people have the same birthday, but when person X arrives there is a match. Find the PMF of X.
For what value of \(c\) is the following function a valid pmf?
\[ p(k) = \begin{cases} \frac{c}{3^k} & k = 1, 2, 3, \dots \\ 0 & \text{otherwise} \end{cases} \]
An airline overbooks a flight, selling more tickets for the flight than there are seats on the plane (figuring that it’s likely that some people won’t show up). The plane has 100 seats, and 110 people have booked the flight. Each person will show up for the flight with probability 0.9, independently. Find the probability that there will be enough seats for everyone who shows up for the flight.
In a chess tournament, n games are being played, independently. Each game ends in a win for one player with probability 0.4 and ends in a draw (tie) with probability 0.6. Find the PMFs of the number of games ending in a draw, and of the number of players whose games end in draws. Hint. If \(D\) is the number of draws, and \(P\) the number of players that draw, then \(P = 2D\).
A fair, six-sided die is rolled forever until all six values have been seen. Compute the expected number of rolls required.
Suppose that a lottery ticket has probability p of being a winning ticket, independently of other tickets. A gambler buys 3 tickets, hoping this will triple the chance of having at least one winning ticket.
What is the distribution of how many of the 3 tickets are winning tickets?
Show that the probability that at least 1 of the 3 tickets is winning is \(3p - 3p^2 + p^3\), in two different ways: by taking the complement of the desired event and applying independence, and by using the pmf of the named distribution from part (a).
Show that the gambler’s chances triple as \(p \to 0\), but that they stay exactly the same as \(p \to 1\).
Evaluate \(\sum_{k = 0}^\infty \frac{k (k - 1)}{2^k}\). Hint. Start with the series \(f(x) = \sum_{k = 0}^\infty x^k\).
Evaluate \(\sum_{k = 0}^\infty \frac{k^2}{2^k}\). Hint. Start with the series \(f(x) = \sum_{k = 0}^\infty x^k\), but do something different than last time.
Evaluate \(\sum_{k = 0}^\infty \frac{1}{3^k (k + 1)}\). Hint. Start with the series \(f(x) = \sum_{k = 0}^\infty x^k\), but do something different than the last two times.