Articles of probability

If I roll two fair dice, the probability that I would get at least one 6 would be…

11 out of 36? I got this by writing down the number of possible outcomes (36) and then counting how many of the pairs had a 6 in them (1,6) (2,6) (3,6) (4,6) (5,6) (6,6) (6,5) (6,4) (6,3) (6,2) (6,1). Is this correct?

Why am I under-counting when calculating the probability of a full house?

I was trying to answer this question. Find the probability of getting a full house from a $52$ card deck. That is, find the probability of picking a pair of cards with the same rank (face value), and a triple with equal rank (different from the rank of the pair of course). My idea was […]

What is the proof that covariance matrices are always semi-definite?

Suppose that we have two different discreet signal vectors of $N^\text{th}$ dimension, namely $\mathbf{x}[i]$ and $\mathbf{y}[i]$, each one having a total of $M$ set of samples/vectors. $\mathbf{x}[m] = [x_{m,1} \,\,\,\,\, x_{m,2} \,\,\,\,\, x_{m,3} \,\,\,\,\, … \,\,\,\,\, x_{m,N}]^\text{T}; \,\,\,\,\,\,\, 1 \leq m \leq M$ $\mathbf{y}[m] = [y_{m,1} \,\,\,\,\, y_{m,2} \,\,\,\,\, y_{m,3} \,\,\,\,\, … \,\,\,\,\, y_{m,N}]^\text{T}; \,\,\,\,\,\,\,\,\, […]

Birthday-coverage problem

I heard an interesting question recently: What is the minimum number of people required to make it more likely than not that all 365 possible birthdays are covered? Monte Carlo simulation suggests 2287 ($\pm 1$, I think). More generally, with $p$ people, what is the probability that for each of the 365 days of the […]

another balls and bins question

I’ve seen many variations of this problem but I can’t find a good, thorough explanation on how to solve it. I’m not just looking for a solution, but a step-by-step explanation on how to derive the solution. So the problem at hand is: You have m balls and n bins. Consider throwing each ball into […]

If a coin toss is observed to come up as heads many times, does that affect the probability of the next toss?

A two-sided coin has just been minted with two different sides (heads and tails). It has never been flipped before. Basic understanding of probability suggests that the probability of flipping heads is .5 and tails is .5. Unexpectedly, you flip the coin a very large number of times and it always lands on heads. Is […]

Logic question: Ant walking a cube

There is a cube and an ant is performing a random walk on the edges where it can select any of the 3 adjoining vertices with equal probability. What is the expected number of steps it needs till it reaches the diagonally opposite vertex?

Prove symmetry of probabilities given random variables are iid and have continuous cdf

Let $Y_1, Y_2, …$ be independent and identically distributed random variables in $(\Omega, \mathscr{F}, \mathbb{P})$ s.t. their distributions are continuous and $$F_{Y}(y) := F_{Y_1}(y) = F_{Y_2}(y) = …$$ For $m = 1, 2, …$ and $i \le m$, define $$A_{i,m} := (\max\{Y_1, Y_2, …, Y_m\} = Y_i), B_m := A_{m,m}$$ Is it really that $$P(B_m) […]

Probability of picking an odd number from the set of naturals?

I know there’s no uniform distribution for a countably infinite set, but I’m wondering if there’s still a way to determine the probability of picking from a subset of a countably infinite set. For example, what’s the probability that I pick an odd number from the set of naturals, assuming I’m picking randomly? Is this […]

Centre in N-sided polygon on circle

What’s the probability that a n-sided polygon made from n distinct random points on circle contain the centre?