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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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Maximizing probability of picking the dime

A money pouch contains a certain number of cents and only one dime. You and your friend are playing a game: They alternate turns and pick one coin at a time, which they put in their pockets. Whoever picks the dime wins. You are trying to decide…
Reyansh Laghari
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Tails of the normal distribution

I'm currently reading Roman Vershynin's High-Dimensional Probability. For Proposition $2.1.2$, I wonder how the lower bound is obtained. I understand that the lower bound is correct, but I don't know where the term $-3x^{-4}$ comes from. Thanks.
Analyst_311419
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Prove the Multiplication Rule (Conditional Form) with more than two events.

Prove the Multiplication Rule (Conditional Form) with more than two events. For events $A_1, A_2,\ldots, A_n$ prove that $$ P(A_1 \cap A_2 \cap\ldots\cap A_n)= P(A_1)\ P(A_2|A_1)\ P(A_3|A_1 \cap A_2)\ \ldots\ P(A_n|A_1 \cap A_2 \cap ... \cap\…
user84324
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Why is the Expected Value different from the number needed for 50% chance of success?

An event with probability $p$ of being success is executed $\frac{1}{p}$ times. For example, if $p=5\%$, the event would then be executed $20$ times. The Expected Value for the total number of trials needed to get one success is $\frac{1}{p}$. In…
user1032613
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Using the law of large numbers to calculate an integral

Let $f:[0,1]\to\mathbb{R}$ be continuous. Prove $$ \lim_{n\to\infty}\int_0^1\int_0^1\cdots\int_0^1f\left((x_1x_2\cdots x_n)^{1/n}\right)dx_1dx_2\cdots dx_n = f\left(\frac1e\right) $$ My idea so far is to use uniform distributions to calculate this.…
Will Harris
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$A$ tosses a fair coin $n+x$ times, $B$ tosses a fair coin $n$ times

This is an extension of the $n+1$ vs n problem here: Probability of $5$ fair coin flips having strictly more heads than $4$ fair coin flips So a common way to think about this is to say that after $n$ tries, both players have the same expected…
narcissa
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Annette, Babette, Colette playing in a $16$ person single elimination tournament

Annette, Babette, Colette, and $13$ other girls are playing in a $16$-player, single-elimination tennis tournament. The $16$ players are placed at random in the first column of the bracket shown in the figure to play $8$ games in Round $1$. The…
Emperor Concerto
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Expected number of coin tosses before $k$ heads

If the probability of getting a head is $p$, how do you compute the expected number of coin tosses to get $k$ heads? I thought this might be the mean of the negative binomial distribution but this gives me $pk/(1-p)$ which is $k$ for $p=1/2$ which…
adalbert
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Finding a rare biased coin from an infinite set

I'm trying to develop an algorithm for finding biased coins. The basic problem formulation is this: There are an infinite number of coins Some proportion $t$ of the coins is biased (this number is known) All biased coins have the same probability…
FowlerA
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Probability puzzle - the 3 cannons

(Apologies if this is the wrong venue to ask such a question, but I don't understand how to arrive at a solution to this math puzzle). Three cannons are fighting each other. Cannon A hits 1/2 of the time. Cannon B hits 1/3 the time. Cannon C hits…
JDS
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Probability of "winning a race"?

Suppose there are three people, A, B, and C. Each person starts at $x=0$ and then randomly one person moves forward by 1, with equal probability among all three people. The winner is the first person to reach $x=3$. By symmetry, the probability of…
user848504
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$ X>0, E(X)=1, E(X^2)=b, \forall a\in (0,1): P(X>a)\geq \frac{(1-a)^2}{b} $

Suppose that $ X>0, E(X)=1, E(X^2)=b $ And We should prove for every $ a $ such that $ 0 < a < 1 $ the following statement: $ P(X>a)\geq \frac{(1-a)^2}{b} $ This is a preliminary course of probability and we learned only basic formulas and…
Yakir
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Probability of A and not B

I'm studying Introduction to probability and currently, I'm stuck with the following problem. Given: $P(A)=0.7$, $P(B)=0.5$, $P(A\cap B)=0.45$ What is the probability of A and not B? I've checked this similar question but I don't understand the…
Dennis
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Random list lengths

I am trying to analyze analytically what would happen if I build a random simulator as follow: I begin with $ N $ lists of length 1, at each iteration, I will pick a random (non-empty) list, reduce its length by 1. And then, with probably $ p $, I…
Andrew Au
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Benford's law and voting in Georgia

With Benford's law "the leading digits of the number found in real-world data sets" should make sense. But this video https://www.youtube.com/watch?v=DoF3WS42w3M&ab_channel=RobertA.Bonavito%2CCPA claims that by applying Benford's law they can prove…
boatcoder
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