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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Why is $ Var(X)= E(Var(X|Y))+Var(E(X|Y)) $?

I see this identity but it does not make sense to me. $ Var(X)= EVar(X|Y)+Var(E(X|Y)) $ Is this something that should intuitively make sense? Is it saying the variance of $X$ is equal to the expected variance of $X$ given $Y$ plus the variance of…
Qwertford
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When to stop in this coin toss game?

Suppose you play the following game: You toss a fair coin. If you get heads, a hundred dollars are added to your reward. If you get tails, however, the game is stopped and you do not get anything at all. After each throw you can decide, whether you…
Ignavia
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Partial marginalization of conditional probability

I was reading about marginalization on Wikipedia, specifically I read: $$p_X(x) = \int_y p_{X\mid Y}(x\mid y)p_Y(y)\,dy$$ I was wondering if the following is true $$\int_y p_{X\mid YZ}(x\mid y,z)p_Y(y) \, dy = p_{X\mid Z}(x\mid z)$$ $X, Y$ and $Z$…
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Randomly picking increasing numbers in $\{ 1, \dots, n\}$

Consider the following procedure (whose input is $N$) that picks increasing numbers from the set $\{ 1, \dots, N \}$ until it picks $N$: i := 0 K_i := 1 while K_i < N pick a number K_{i+1} from the set { K_i, ..., N } uniformly at random i =…
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Why are events $A$ and $B$ independent whilst events $A$ and $C$ are not?

Two ordinary fair dice (one red and one blue) are thrown. Event A: The red die will show a 5 or a 6. Event B: The sum of the two dice will be 7. Event C: The sum of the two dice will be 8. Using the test for independence: $$P(A\cap B) = P(A).P(B) $$…
Kantura
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Normal variables - adding and multiplying by constant

$X\sim N(a,b)$, while $c$ is constant. Is is true that then: $X+c\sim N(a+c,b)$ ? $cX\sim N(c\cdot a, b)$ ?
Happy man
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What is the probability of shuffling a deck and not getting a repeating card for the whole deck?

What are the odds of shuffling a deck of 52 cards and going through each card one by one without getting any repeats (like 2 kings back to back). I'm bad at this kind of deduction, but at a high level it seems roughly like 1 in 13 odds over 52…
Maxx
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What is the smallest and largest possible values for the variance?

Suppose $P( X \in \{1,2,3\}) = 1$ and $E(X) =2.5.$ What is the smallest and largest possible values for the variance? My understand: So what I understand is variance finds the distance between each element and the mean. So the closer 2 is to E(X)…
Javant
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Expected number of rolls to get 1 followed by 2

We have a fair, six-sided dice. The questions are What's the expected number of rolls to get 1 followed by 1? What's the expected number of rolls to get 1 followed by 2? Let $E$ be the expected time to get '11'. From the geometric distribution,…
user369210
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Mean distance between a fixed point and a gaussian distributed random variable

In 2D or 3D I have a fix point y and Gaussian distribution of a random point x. I am now interested in the mean euclidean distance between x and y: $E_x [ d(x,y) ] = \int _{-\infty }^{\infty } d(x,y) N(x; \mu, \Lambda )dx$ Many thanks in advance
Matthias
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N points on a Circle

Given n points drawn randomly on the circumference of a circle, what is the probability they will all be within any common semicircle?
Benji
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What is the probability that the coin drawn is fair

A box contains $n$ number of coins, $m$ of which are fair and the rest are biased. The probability of getting a head when a fair coin is tossed is $1/2$, while it is $2/3$ when a biased coin is tossed. A coin is drawn from the box at random and is…
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A "trick" with a deck of cards, probability of equal result

I recalled this "trick" I read in some magazine years ago and thought I'd try to explore it a bit more. You have standard deck of 52 cards. The cards are on the table, facing up. Your friend secredly selects a number, $n_0$, from 1 to 13. Then your…
Valtteri
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probability question related to pattern in coin tossing

If I toss a fair coin $n$ times, calculate the probability that no pattern HHTHTHH occurs.
Qiang Li
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Probability of having a Girl

A and B are married. They have two kids. One of them is a girl. What is the probability that the other kid is also a girl? Someone says $\frac{1}{2}$, someone says $\frac{1}{3}$. Which is correct? Now A and B have 4 children and all of them are…
Hailey
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