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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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How to prove Boole’s inequality

I am trying to prove Boole’s inequality $$P\left(\ \bigcup_{i=1}^\infty A_i\right) \leq \sum_{i=1}^\infty P(A_i).$$ I can show it of any finite $n$ using induction. What to do for $\infty$ ?
Ashot
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probability of getting the same number of tails

Alice tosses a fair coin $n$ independent times and Bob tosses a fair coin $m$ independent times. Find an elegant or clever argument to compute the probability that they have equal numbers of Tails. It had better not involve any lengthy sums. The…
tostito
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"Pairwise independent" is weaker that "independent"

Can someone please give me a reference to an (simple, realworld, i.e. not constructed) example of a discrete probability space such that there are three events in it that are pairwise independent but all three together are not independent (although…
MyCatsHat
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I have a bag with 3 coins in it. One of them is a fair coin, but the others are biased trick coins.

When flipped, the three coins come up heads with probability 0.5, 0.3, 0.6 respectively. Suppose that I pick one of these three coins entirely at random and flip it three times. 1. What is P(HTT)? (i.e., it comes up heads on the first flip and tails…
White Mamba
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Probability of a nonnegative integer-valued random variable being zero

I'm trying to teach myself probability theory, and an exercise has me stumped. This exercise comes from Alon & Spencer 4.8, in the chapter on the second moment method. Let $X$ be a random variable taking values in $\mathbb{Z}_{\geq 0}$. Let $E[X^2]$…
JeremyKun
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Why $\frac{X+YZ}{\sqrt{1+Z^2}} \sim \mathcal{N}(0,1)$ if $X,Y,Z$ are i.i.d. $\mathcal{N}(0,1)$?

The original question was: Suppose $X,Y,Z$ are i.i.d. $\mathcal{N}(0,1)$, find a nonnegative continuous function $g$ such that $\frac{X+YZ}{g(Z)} \sim \mathcal{N}(0,1)$. The solution says, since $E(X+YZ\mid Z=z)=0$ and $Var(X+YZ\mid Z=z)=1+z^2$, for…
David Tan
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Joint Bernoulli Distribution

If $X$ and $Y$ are two (not necessarily independent) Bernoulli's with success probabilities $a$ and $b$ resp., how do we construct the joint dist. in terms of $a$,$b$, and $\rho$---the correlation? I can get $\mathbb{P}(X=1,Y=1)$ by manipulating…
Peter
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Expectation of joint life span

The life span of a particular mechanical part is a random variable described by the following PDF: If three such parts are put into service independently at t=0, determine a simle expression for the expected value of the time until the majority of…
qed
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Why is that the events (Sum of dice roll=6, first die=4) are dependent, but the events (Sum of dice roll=7, first die =4) are independent?

Roll $2$ dice. Let $E$ be the event that the sum of the dice is $6$ Let $F$ be the event that the sum of the dice is $7$ Let $G$ be the event that the first die rolled is a $4$ $E$ and $G$ are dependent (since $P(E\cap G) \neq P(E)P(G)$ ) $F$ and…
merkuryblast3r
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A coupling card trick in Durrett's book

This is an example in Durrett's book "Probability theory: theory and examples", it's about the coupling time in Markov chains, but I can't see the reason behind it. The trick is played by two persons A and B. A writes 100 digits from 0-9 randomly, B…
zemora
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Is the chance of breaking even in this coin toss game $43.75\%$?

I'm trying to do the maths on a coin toss game after 100 games but think I am failing. The rules are as follows. we start with 1000 coins we always bet that heads come up minimum bet is 10 coins maximum bet is 80 coins if tails comes up, we lose…
Peter Bushnell
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Find probability of biased Coins

There are two biased coins A and B. We have been given the following: P(H|Coin A) = 0.9 P(T|Coin A) = 0.1 P(H|Coin B) = 0.1 P(T|Coin B) = 0.9 We are also given that Probability of tossing Coin A or Coin B randomly is 0.5 How could we find the…
user3036345
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The hot hand and coin flips after a sequence of heads

ESPN recently posted a story demonstrating that the "hot hand" concept is, in fact, real. Part of the justification is this example based on coin flips from a paper by Adam Sanjurjo and Joshua B. Miller: And now [Joshua] Miller brings it back to…
Barry
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Probability that these characters win a game.

I have 11 characters, $[2,3,4,5,6,7,8,9,10,11,12]$, and they all play a game. Game Description: All players stand at the start line $n$ spaces away from finish line. Two fair dice are rolled. The two results on each of the dice are summed up, and…
Xetrov
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Probability that $\sin\theta \cos\phi < 0.999772$

I am solving a kinematics question in particle scattering. The final answers lies in finding the probability such that $$\sin\theta \cos\phi < 0.999772$$ The ranges of $\theta$ and $\phi$ are $0\leq\theta<\pi$ and $0\leq\phi<2\pi$.
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