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Questions tagged [probability]

For questions about probability. independence, total probability and conditional probability. For questions about the theoretical footing of probability use [tag:probability-theory]. For questions about specific probability distributions, use [tag:probability-distributions].

The probability that an event occurs is a number in the interval $[0, 1]$, which represents how likely the event is to happen. $0$ indicates it will never happen, $1$ indicates it will always happen.

For example, throwing two dice gives a total of $6$ five times out of thirty-six. We write $$P(X=6)=\frac{5}{36}$$.

Use this tag for basic questions about probability, independence, total probability and conditional probability.

For questions about the theory of probability, use instead. For questions about specific probability distributions, use .

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A complicated problem on probabilistic conditioning

The real random variables $X$ and $Y$ are independent and both have a Poisson distribution with the parameter 1, i.e. Po(1). Find: $$\mathbb{E}\left[ \left( 2^{2X}+2^{Y} \right)^2|X+Y \right]$$ Answer: $$\left(\frac{9}{2}\right)^{X+Y} +2\cdot…
Vadim Omelchenko
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Joint probabilities, conditional probabilities with the chain rule.

I'm reading through a book, and it walks through a problem. We need to compute $p(a | e, f)$. It says that by applying the chain rule we can see: $$p(a|e,f) = \frac{p(e,a|f)}{p(e|f)}$$ Looking at the chain rule, I do not understand how that was…
oadams
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A colored ball problem

Say you have $2n+2b$ balls where $2n$ balls are colored white, $b$ balls are colored blue and $b$ balls are colored red. You have two urns. You randomly choose $n+b$ balls and throw in urn $1$ while you place the remaining $n+b$ balls in urn $2$.…
user257494
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Probability of Winning at least 7 times

Tried to model a popular game I was playing, but the probabilities seemed off. A game allows you to have up to 12 wins but only allows 3 losses. Each win/lose is independent from each other with a 50% probability and assuming we play until 12 wins…
Mid
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In a game of Bridge, what is the probability that all 4 players are dealt 13 cards of the same suit?

I was asked this question by a student at my college, and I answered it like this: Since Bridge is played with 4 players, and there are 4 suits per deck of 52 cards, and assuming the deck is a fair, properly shuffled deck of cards, then the…
Jabernet
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What is the probability that a customer will not use a credit card? Pays in cash or with a credit card?

So I'm doing some basic probability problems for homework, and we just recently went over the Inclusion-Exclusion prinicple, which I'm assuming this problem deals with, which is as follows. Shoppers can pay for their purchases with cash, a credit…
secondubly
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Given two sets of $100$ samples of $10$ items from a $1000$ item set, what is probability that the two sets have non-empty intersection

Suppose two people go grocery shopping $100$ times each. Each time, they pick $10$ items randomly from the $1000$ items at the store. As a result, each person has $100$ randomly chosen baskets of $10$ items. What is the probability that, by the…
user83387
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Prove that if $E[X|\sigma(Y)] = Y$ and $E[Y|\sigma(X)] = X$ then $X = Y$ almost surely.

Prove that if $E[X|\sigma(Y)] = Y$ and $E[Y|\sigma(X)] = X$ then $X = Y$ almost surely. This is my idea: By assumption, $Y = E[X|\sigma(Y)] = E\left\lbrack E[Y|\sigma(X)]|\sigma(Y)\right\rbrack$, and I would like to show that this equals…
Ann Marie
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What is the probability that Raj and Rana have atleast 3 persons between them,given raj and rana are standing in a row and there are 9 peoples.

Question: Raj and Rana are standing in a row. There are 9 persons including Raj and Rana. What is the probability that at least 3 people will stand between Raj and Rana? My solution: $9$ people can be arranged in $9!$ ways. Excluding Raj and Rana,…
Jack
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Prove that $\mathrm E\left({\max}\left\{X^2 , Y^2\right\} \right) \leq 1 + \sqrt{1 - \rho^2}$ where $\rho$ is their correlation.

I have 2 random variables $X,Y$ with mean 0 and variance 1, their correlation is $\rho$. I need to prove this inequality $$\mathrm E\left({\max}\left\{X^2 , Y^2\right\} \right) \leq 1 + \sqrt{1 - \rho^2}$$ I need some pointers as to how to solve…
SphericalCow
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Past coin tosses affect the latest one if you know about them?

Suppose Mark and Paul are sitting on a table, and Mark starts tossing an unbiased and fair coin. He tosses it for 99 times, and he gets 99 consecutive tails. At this point Mark asks Paul: "let's bet $100 on the next toss, do you want to pick tail or…
Daniel Scocco
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Calculating probability for forming a triangle

I am having trouble coming up with a solution for this problem: There is a stick of unit length. We break it into two parts. Now, we pick the bigger one and break it into two parts. I want to calculate the probability that the three pieces form…
Мартин Радев
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What is the probability of an event happening in some interval given probability of it in x interval?

Suppose there is an event that happens with a probability of y in x interval of time, what would be the probability of it happening in x/2 interval of time? Would that be y/2 or is there something more to be considered? To be a bit more specific,…
Nishant
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Coin flipping game with stop-loss

You play 100 rounds of a coin flipping game where you win \$2 for a head and lose \$1 for a tail on each round. Clearly since the coin tosses are independent the expected winnings are \$50. Now, suppose you play at most 100 rounds of this game as…
ScarletPumpernickel
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Conditional expectation on Gaussian random variables

If we suppose that the two independent random variables $X \sim \mathcal{N}(0,\sigma^2_x)$ and $N \sim \mathcal{N}(0,\sigma^2_n)$ and that $S = X + N$, how would I work out the conditional expectation $E[X\mid S=s]$?
John Lee
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