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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 .

105859 questions
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Flip a coin, but you lose when tails appears

I have the following game: you flip a coin (heads with probability $p$), and if you get heads you earn $1000$ dollars, and you can decide if you want to flip again. If not, you keep the money. However, if you get tails, the game is over and you go…
FifteenPointOne
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How to show that $|P(A\cap B)−P(A)P(B)|≤\frac{1}{4}$?

I am trying to show that $$ |P(A\cap B)−P(A)P(B)|≤\frac{1}{4} $$ But I don't see how to begin... If $A$ and $B$ are independent the result is obvious. Can someone give me a hint?
Michelines
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What is the expected number of coin tosses it would take to get N many heads OR N many tails?

Where we do NOT require that the heads or tails be consecutive (though they may be!) Obviously, this expectation, $E[T]$, is bound as follows: $N < E[T] < 2N - 1$ And obviously $E[T] = \sum_{i=N}^{n=2N-1}i*P[Game \ Ends \ On \ i^{th} \ Round]$,…
TheMathBoi
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Is the probability of at least $k$ consecutive heads higher for a coin with higher probability of heads?

Suppose a coin has probability $p$ for heads and $(1-p)$ for tail. Let $P_{k,p}$ be the probability that in $N$ flips there is a sequence of consecutive heads of length greater than or equal to $k$. $N$ is some fixed number greater than $k$. Does…
Tom Mosher
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Variance of Binomial Distribution $E[X^2]$

so I'm trying to use the equation: $Var(X) = E[X^2] - (E[x])^2$, And for the $E[X^2]$ part, I'm trying to use the method of indicators... However, when I do that, I get the same value as with $E[X]$... Is it wrong to try to use the method of…
Joy
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Dice rolling probability game

In a hockey-themed board game, players start the game in the penalty box. If rolling the same number on both dice is required to escape from the penalty box, and Piper, Quincy, and Riley take turns, in he order named, rolling a pair of standard six…
Max0815
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Why is my explanation wrong?

A class of $30$ students, J. being one of them, has $5$ classes today. What is the chance, that J. will be the chosen student to explain the homework in at least two classes? What is the chance that someone will be chosen at least twice? My…
DaniFoldi
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Elementary proof that a binomial distribution with $p\ne \frac12$ is almost symmetric?

I am looking for an elementary explanation of why a loaded coin gets more and less than the expected number of heads approximately equiprobably, in other words, mean=median. Mathematically speaking, if $X\sim B(n,p)$ is a Binomial random variable,…
sds
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Probability of picking a specific value from a countably infinite set

I have just learned in probability that picking a specific value from an uncountably infinite set (continuous) has a probability of zero, and that we thus estimate such things over an interval using integrals. This clearly does not apply to finite…
user9034
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Randomly selected subset, expected value of the sum

Interesting problem I spotted while learning: Let $X=\left\{1,..,n\right\}$. We randomly select subset of $X$ and name it $A$. Each subset if equally likely. a) Find the expected value of the sum of elements of A. b) Find the expected value of the…
xan
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Expected value of largest integer in a draw

Suppose I pick $k$ integers without replacement from $\{1, \ldots, n\}$. Let $I$ be the value of the highest integer. A calculation with binomials reveals $$E[I] = \frac{k}{k+1}(n+1)$$ This is a very simple formula - does it have a simple…
Michael S.
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Urn and probabiblity

Two urns contain n balls each, numbered from 1 to n. We pick a ball from the first one and then a ball from the second. What is the probability that the number of the second ball is a) smaller b) equal to the number of the first ball? My humble…
Samuel
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3 coins - 1 biased

We have 3 coins, of which 2 are fair and the 3rd is biased (gives heads with probability 4/7). Two friends throw, in turn, all coins together, one time each, and the winner is the one who gets more heads. What is the probability of a draw? My first…
Chen Aavaz
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Rolling two dices n-times. Probabillity of getting doubles

I started learning the basics of probabillity theory by myself and did some practicing and I was doing the following exercise which i found on the internet. Two dices are rolled n-times, determine the probabillity of getting each double…
johnka
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Bayes' Theorem in Stephen Baxter's Manifold: Time

I am currently reading the sci-fi novel Manifold: Time by Stephen Baxter, which contains the following problem. You are given a box which has either 10 marbles or 1000 marbles. By pressing a lever on the box, one marble is randomly taken out and…
user55308
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