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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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Show that $P(X+Y+Z\text{ is a multiple of }3)\ge 1/4$

Suppose a box contains tickets, each labeled by an integer. Let $X,Y$ and $Z$ be the results of draws at random with replacement from thw box. Show that no matter what the distribution of numbers in the box, $$P(X+Y+Z\text{ is a multiple of }3)\ge…
caligirl11
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Relative entropy is non-negative

Let $p=(p_1,\dotsc,p_r), q=(q_1,\dotsc,q_r)$ be two different probability distributions. Define the relative entropy $$h(p||q) = \sum_{i=1}^r p_i (\ln p_i - \ln q_i)$$ Show $h(p||q)\geq 0$. I'm given the hint that I should show $-x\ln x$ is concave…
anon
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A multinomial problem (balls, bins, etc.)

Consider the well known multinomial setting: there are L balls, thrown at random at n bins so that the probability that a ball falls in bin i is $p_i$, independent of the other balls (the $p_i$’s are all positive and sum to 1). Let $X_i$ be the…
Buchuck
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The definition of independent discrete random variables

In probability books, the definition of independent discrete random variables are often given as The random variables $X$ and $Y$ are said to be independent if $\mathbb P(X \leq x, Y \leq y) = \mathbb P(X \leq x) \mathbb P(Y \leq y)$ for any…
NonalcoholicBeer
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PDF of a sum of exponential random variables

Let $X_i$ for $i=1,2,...$ be a sequence of i.i.d exponential random variables with common parameter $\lambda$. Let $N$ be a geometric random variable with parameter $p$ that is independent of the sequence $X_i$. What is the pdf of the random…
Wintermute
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Produce output with certain probability using fair coin flips

"Introduction to Algorithm" C.2-6 Describe a procedure that takes as input two integers a and b such that $0 < a < b$ and, using fair coin flips, produces as output heads with probability $a / b$ and tails with probability $(b - a) /b$. Give a…
NonalcoholicBeer
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What are the odds of getting heads 7 times in a row in 40 tries of flipping a coin?

I know if you flip a coin $7$ times, the odds of getting $7$ heads in a row is $1$ in $2^7$ or $1$ in $128$. But if you flip a coin $40$ times, what are the odds of getting $7$ heads in a row in those $40$ tries? I only want to know the first time…
Coin Flipper
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understanding Log likelihood

I have always had a problem of understanding the big picture of probability and get lost every time I am studying it. I am working on implementing a X-means algorithm for clustering data and the log likelihood function keeps popping up. Please can…
persistence911
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Thinning a Poisson Process

Suppose that events are produced according to a Poisson process with an average of lambda events per minute. Each event has a probability $p$ of being Type A event, independent of other events. Let the random variable $Y$ represent the number of…
statStudent
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Conditional distribution of binomial random variables is hypergeometric

Let's say $X$ and $Y$ are binomial random variables with parameters $n$ and $p$ and $X+Y=m$. I want to show that the conditional distribution of $X$ if $X+Y=m$ is a hypergeometric distribution. I'm thinking about putting these in terms of coin flips…
user108431
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Proving an upper bound for Prob[X>=E[X]]

Let random variable $X\sim\text{Binomial}\left(a+b,\frac{a}{a+b}\right)$, where $a$ and $b$ are positive integers. I'm trying to prove that $\mathbb{P}[X\geq\mathbb{E}[X]]\leq\frac{3}{4}$, which appears to be true numerically. Does anyone have a…
Marcus Fry
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Probability that $A$ need more coin tosses to get two consecutive heads than $B$ need to get three consecutive heads

Two people $A$ and $B$ throw fair coins independently. Let $M$ be the number of coin tosses until $A$ gets two consecutive heads. Let $N$ be the number of coin tosses until $B$ gets three consecutive heads. What is the probability that $M>N$?
minimax
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Probability of choosing the same number

Assume $n$ people choose a number between $1$ and $k$ uniformly at random, simultaneously. What is the probability that any two of the $n$ people get the same number? I tried: The probability that two people choose the same number is $\frac1k$.…
sgp
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Probability of Defective coins

Among 100 coins one is defective: it has two heads. One chooses a coin (a good or bad one) and tosses it 10 times. It turns out that the head comes out all 10 times. What is the probability that the head comes out again when the coin is tossed one…
mike
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Expectation of an event

Let $A$ be an array of length 1000 with all entries 0. I want to fill up $A$ with ones using the following approach: At each iteration I take three random integers $(j_1,j_2,j_3)$ from [1,1000] with replacement and do the following: Set…
user12290
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