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) […]

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 $\mathrm{cov}( f(X),g(X)) \geq 0$ 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?

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 […]

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 […]

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 $p$ […]

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 […]

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 […]

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 […]

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.

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