Articles of conditional probability

Conditional probabilties in the OR-gate $T=A\cdot B$ with zero-probabilities in $A$ and $B$?

My earlier question became too long so succintly: What are $P(T|A)=P(T\cap A)/P(A)$ and $P(T|B)=P(T\cap B)/P(B)$ if $P(A)=0$ and $P(B)=0$? I think they are undefined because of the division by zero. How can I specify the conditional probabilities now? Please, note that the basic events $A$ and $B$ depend on $T$ because $T$ consists of them, […]

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 […]

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\}} + […]

Conditioning on a random variable

The number of storms in the upcoming rainy season is Poisson distributed but with a parameter value that is uniformly distributed between (0,5). That is Λ is uniformly distributed over (0,5), and given Λ = λ, the number of storms is Poisson with mean λ. Find the probability there are at least three storms this […]

Conditional probability for an exponential variable

Let $X$ be an exponential random variable with parameter $\lambda$. I have to find $\mathbb{E}[X\mid X<\alpha]$ where $\alpha >0$. I must find $\mathbb{P}(X\mid X<\alpha)$ first. What I did is as follows: $\mathbb{P}(X\mid X<\alpha)=\dfrac{\mathbb{P}(X<t \;\text{and}\; X<\alpha)}{\mathbb{P}(X<\alpha)}$, where I assumed $t> \alpha$ then I get the conditional probability to be equal to $1$. I’m not sure if […]

Brownian Motion Conditional Expectation Question

I have a real number $x$, and $W$ is a standard Brownian motion. Let $0 < s < t$. How to find $$ \mathsf E[W_s | W_t = x] $$ Please provide me with a step by step answer as I want to understand your steps and the concept. Many thanks in advance.

Probability Bertsekas Question

Question: We have two jars each containing an equal number of balls. We perform four successive ball exchanges. In each exchange, we pick simultaneously and at random a ball from each jar and move it to the other jar. What is the probability that at the end of the four exchanges all the balls will […]

Show: $\mathbb{E}(f|\mathcal{F})=\mathbb{E}(f)$

Let $(\Omega,\mathcal{A},\mathbb{P})$ be a probability space and $\mathcal{F}$ a sub-$\sigma$-algebra. Let $\mathcal{F}$ be trivial, i.e. $\forall A\in\mathcal{F}: \mathbb{P}(A)\in\left\{0,1\right\}$. Show that $\mathbb{E}(f|\mathcal{F})=\mathbb{E}(f)$. One criterion to prove that is to show that $$ \forall A\in\mathcal{F}: \int_A\mathbb{E}(f)\, d\mathbb{P}=\int_Af\, d\mathbb{P}. $$ Do not know exactly how to show that in common. For the special case, that $\mathcal{F}=\left\{\Omega,\emptyset\right\}$ it is […]

Secret santa probability

26 people put their name in a hat. Each person picks a name out of the hat to buy a gift for. If a person picks out themselves they put the name back into the hat What is the probability of the last person picking themselves? I only ask this because it just happened!!

Conditional Expectation of random sum of independent random variables

Let $Y=X_1+X_2+\dots+X_N$ where $X_1,X_2,\dots,N$ are jointly independent random variables, $X_1,X_2 …$ identically distributed continuous random variables with finite expectation, and $N$ a discrete random variable with finite expectation. What is the definition of conditional density $f(y|n)$ of $Y$ given $N=n$ (if it exists). I need it to compute $E(Y|N=n)$. Is there another approach to compute […]