Articles of probability

Mutually Exclusive Events (or not)

Suppose that P(A) = 0.42, P(B) = 0.38 and P(A U B) = 0.70. Are A and B mutually exclusive? Explain your answer. Now from what I gather, mutually exclusive events are those that are not dependent upon one another, correct? If that’s the case then they are not mutually exclusive since P(A) + P(B) […]

What is the probability of two dice getting a sum of 7 without a two?

I am currently working on conditional probability and I am somewhat confused about how exactly to complete this problem. I know that to find conditional probability that you utilize: $$P(A|B) = \frac{P(A\cap B)}{P(B)}$$ I also know that there is a $6/36$ chance to roll a sum of 7, and that if you roll a sum […]

How to show $f(X)$ and $g(X)$ are positively correlated for a random variable $X$, if $f$ and $g$ are non-decreasing and bounded?

How to show $\mathrm{cov}( f(X),g(X)) \geq 0$ for a random variable $X$, if $f$ and $g$ are non-decreasing and bounded? I know a coupling method by introducing an independent copy of $X$ (page $2$, here). What’s the straightforward way of showing this?

Markov Chain and Forward and Backward Probabilities with Alice and Bob

System Alice and Bob are moving independently from one city to another. There are $d$ cities, the probability of moving to another city (for each individual) is $m$ and each move is equiprobable (there is no preferred city). The choice of moving and choice of where to move to of Alice are independent of the […]

Weird equality of expectations involving stochastic integral

First of all, sry for the title. I just couldn’t figure out any better description for this weird problem: Let $X$ be a bounded real r.v. and $(A_t)_{t\geq0}$ an increasing bounded process (hence $A_{\infty}\leq M$ a.s for some real $M$.). EDIT: And assume additionally that $A(0)=0$ a.s. Show that $\mathbb{E}[XA_{\infty}]=\mathbb{E}[\int_{0}^{\infty}\mathbb{E}[X|\mathcal{F}_t]\mathbb{d}A_t]$ Since this exercise is really […]

Sum of Binomial random variable CDF

Suppose there are two independent Binomial random variables $$ X\sim Binomial(n,p)\\ Y\sim Binomial(n,p+\delta) $$ where $\delta$ is considered to be fixed, and $p$ can vary in $(0,1-\delta)$. Now consider the following value (sum of two probabilities): $$ \mathbb{P} \left( X\geq(p+\frac{\delta}2)\cdot n \right) +\mathbb{P} \left( Y\leq(p+\frac{\delta}2)\cdot n \right) $$ My question is: Which value of $p$ […]

Large variance implies small probability of being in interval

Let $X$ be some random variable on a real interval $[0,\infty)$. Let $[a,b]\subset [0,\infty)$. Intuitively, it seems reasonable that if the variance of $X$ denoted by $\sigma_X^2$ is large compared to the size of the subset $b-a$, then the probability of being inside the subset should be small (if it was high then the variance […]

Subset Probability to Element Probability

Is there any way to match (or map) from Subset Propabilities to Element Probabilities? Suppose that John may select x-sized subsets from a population of N items. In every subset he has exactly x items. Let subset A has for example {item1, item3, item7} and subset B has {item2, item3, item9}. How can I compute […]

Betting a constant fraction on a biased coin

I’m looking at a betting game where I have \$100 and want to double my money by repeatedly betting on a biased coin; it shows heads with probability $p<\frac{1}{2}$ in which case I win even money. I imagine my best strategy is to go all-in, but I also wanted to investigate what happens if I […]

NBA round robin probability

Let’s say there is a team who you expect to win 75% of its games in a given 82-game NBA regular season (and the probability of winning each game = 75%). What is the probability that the team will never lose consecutive games at any point during the 82-game season.