Articles of probability

Given K balls and N buckets what is the expected number of occupied buckets

Given K balls and N buckets how do you calculate the expected number of buckets with at least 1 ball. Each ball is put in a bucket chosen at random with a uniform probability distribution. Assume also K $\leq$ N.

Probability Question – An elevator & 5 Passengers

I’m working on this problem: An elevator in a building starts with 5 passengers and stops at seven 7 floors. If each passenger is equally likely to get an any floor and all the passengers leave independently of each other, what is the probability that no two passengers will get off at the same floor? […]

Flip two coins, if at least one is heads, what is the probability of both being heads?

Quick basic question here to make sure I understand conditional probability properly. You flip two coins, and at least one of them is heads. What is the probability that they are both heads? Now, I think the answer to this is $\frac{1}{3}$, for the following explanation. If $A$ is the event that the first coin […]

Andrei flips a coin over and over again until he gets a tail followed by a head, then he quits. What is the expected number of coin flips?

(a) Andrei flips a coin over and over again until he gets a tail followed by a head, then he quits. What is the expected number of coin flips? (b) Bela flips a fair coin over and over again until she gets two tails in a row, then she quits. What is the expected number […]

Are there any random variables so that E and E exist but E doesn't?

Are there any random variables so that E[X] and E[Y] exist but E[XY] doesn’t?

Pick a card from a set with the help of a dice

Some days ago i was playing “Settlers of Catan”, an homemade version. In this game there is an event where a players pick, randomly, a card from another player. So here is the problem: can we use the dice (six faces) to pick the card randomly? Until the opposite player has less than seven cards, […]

What is the expected time to cover $m \leq N$ elements in a set by sampling uniformly and with replacement?

The expectation for covering a set of $N$ elements by sampling uniformly and with replacement is $N * H(N)$, where $H(N)$ is the $N$th harmonic number. Is there a similar expression using harmonic numbers for the expectation for the number of sampling events necessary to see any $m \leq N$ unique elements in the set? […]

Average number of trials until drawing $k$ red balls out of a box with $m$ blue and $n$ red balls

A box has $m$ blue balls and $n$ red balls. You are randomly drawing a ball from the box one by one until drawing $k$ red balls ($k < n$)? What would be the average number of trials needed? To me, the solution seems to be $$\sum_i i * \frac{\mbox{the number of cases where k-th […]

Is the Event a Conditional Probability or an Intersection?

My question is based on Example 1.9, p 22, *Introduction to Probability (1 Ed, 2002) by Bertsekas, Tsitsiklis. Define the event $A$ = {an aircraft is present} and $R$ = {the radar registers an aircraft presence}. Express the following events in terms of $A$, $R$, and/or their complements. $\begin{align} & \text{(i) The radar correctly registers […]

Conditional expectation of an uniformly distributed random variable

Suppose $U_1, \ldots, U_n$ are i.i.d. random variables with $U_1$ distributed uniformly on the interval $(-1, 1)$. Compute $\mathbb{E}(U_1 + \ldots + U_n |\max(U_1, \ldots, U_n) = t)$ for $t \in (-1, 1)$. So my attempt was to try to compute $\mathbb{E}(U_1 |\max(U_1, \ldots, U_n) = t)$, writing $U_1 = U_1 1_{\{U_1 < t\}} + […]