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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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the expectation of a chocolate bar

So my buddy claims that if I split a chocolate bar at random into two pieces, then the expected size of the larger piece is $\frac{3}{4}$ of the bar. I can't figure out how he came up with this value... Can someone explain this? If you can, can you…
Hristo
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Probability that the sum of 'n' positive numbers less than 2 is less than 2

I'm a high school student, And I stumbled across this problem Q) if you arbitrarily choose 3 real positive numbers less than or equal to 2 What is the probability that their sum is less than or equal to 2? In my course I have learnt to solve 2…
amay_varma
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A samurai cuts a piece of bamboo

Suppose a samurai wants to try out his new sword and cuts a piece of bamboo twice, randomly, so now there are $3$ lenghts of bamboo. What is the probability of these 3 pieces being able to form a triangle? I have never came across a continuous…
Leonardo Fontoura
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How to prove Bonferroni inequalities?

Define $$S_1 = \sum_{i=1}^n P(A_i)$$ and $$S_2 =\sum_{1 \le i < j \le n}^n P(A_i \cap A_j)$$ as well as $$S_k =\sum_{1 \le i_1 < \cdots < i_k \le n}^n P(A_{i_1} \cap \cdots \cap A_{i_k})$$ Then for odd $k$ in…
LittleSweet
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Is this lot drawing fair?

Sorry for a stupid question, but it is bugging me a lot. Let's say there are $30$ classmates in my class and one of us has to clean the classroom. No one wants to do that. So we decided to draw a lot - thirty pieces of paper in a hat, one of which…
brilliant
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Odds of Winning the Lottery Using the Same Numbers Repeatedly Better/Worse?

Does the probability of winning the lottery differ between randomly generated numbers vs. selecting the same numbers every time? Specifically. I'm interested in a breakdown of the odds per number for a given set of numbers that comprise a single US…
afilbert
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A king has n children, at least one of them is a daughter. What’s the probability that all of them are daughters?

A king has n children, at least one of them is a daughter. What’s the probability that all of them are daughters? So far I've considered the case of 3 children, which gives a probability of 1/7. But I'm confused about how to generalise this?
claud124
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Expectation of CDF of continuous random variable $X$, evaluated at $X$

Given the continuous random variable $X$ with cumulative distribution function $F_{X}$, find $E[F_{X}(X)]$. Attempt at solution: I understand that the expected value, $E[X]$, of a random variable, $X$, is $\int^{+\infty}_{-\infty} x…
Swamp G
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Is this always true: $P(A|B) = 1-P(A^c|B)$?

Does this identity hold for all events? $$ P(A|B) = 1-P(A'|B) $$ Logically speaking, if the probability of $A$ given $B$ occurred is $X$, shouldn't the probability that $A$ does not occur, $A'$, given $B$, be similarly $1-X$? There is a related…
Plastic Astronaut
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Proof of the inclusion-exclusion formula in probability

Let $(\Omega,\mathcal F, P)$ be a probability space and let $A_1,A_2,...,A_n$ be events in $\mathcal F$. Prove the following inclusion-exclusion formula $P(\bigcup_{i=1}^nA_i)=\sum_{k=1}^n$ $\sum_{\mathcal J \subset \{1,...,n\}; |\mathcal J|=k}…
user100106
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Given n ranging from $1$ to $100$, find sum of digits equal to half of arithmetic sum of $1$ to $100$

I have a number sequence from $1$ to $100$. Given $2$ bins, the numbers are randomly assigned to each bin. I know the total sum from $1$ to $100$ is $5050$. Thus, for both bins to have the same sum, each bin must sum up to $2525$. All the $100$…
lone
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Probability that two people see each other at the coffee shop

Two mathematicians each come into a coffee shop at a random time between 8:00 a.m. and 9:00 a.m. each day. Each orders a cup of coffee then sits at a table, reading a newspaper for 20 minutes before leaving to go to work. On any day, what is the…
Sophie Alpert
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Probability of rain after an amount of time in Minecraft game

Minecraft time is measured in ticks. When a world is loaded, the game waits anywhere from 12,000 to 180,000 ticks to start raining. After it starts raining, the rain lasts anywhere from 12,000 to 24,000 ticks. When the rain ends, clear weather lasts…
Benjamin Gilbert
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Expected value of game involving 100-sided die

The following question is from a Jane Street interview. You are given a 100-sided die. After you roll once, you can choose to either get paid the dollar amount of that roll OR pay one dollar for one more roll. What is the expected value of the…
90b56587
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Regression towards the mean v/s the Gambler's fallacy

Suppose you toss a (fair) coin 9 times, and get heads on all of them. Wouldn't the probability of getting a tails increase from 50/50 due to regression towards the mean? I know that that shouldn't happen, as the tosses are independent event.…
Joebevo
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