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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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Probability on spreading of rumors

A little help here. Exercise 21, Ch. 2 from Feller's book reads In a town a $n+1$ inhabitants, a person tells a rumor to a second person, who in turn repeats it to a third person, etc. At each step, the recipient of the rumor is chosen at random…
r_31415
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E[XY] from table

This is an example from a book that I dn't really understand. X=1 | X=2 Y=3 | 0.3 | 0.1 Y=6 | 0.1 | 0.5 $$E(XY)=\sum_{all\;y}\sum_{all\;x}xyp_{x,y}(x,y)=8.1$$ I can't grasp how this dubble sum works. I thought it was something lke this: First…
Olof
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Coin sequence paradox from Martin Gardner's book

"An event less frequent in the long run is likely to happen before a more frequent event!" How can I show that THTH is more likely to turn up before HTHH with a probability of 9/14, even though waiting time of THTH is 20 and HTHH, 18! I would be…
KH Kim
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$\lim_{n \to \infty} {\mathbb E}X_n$ for a coin flipping payoff problem

Suppose we have a fair coin and we start with a base amount of money $X_0 = C \in {\mathbb N}$, and each time we flip the coin we have $X_{n+1} = X_n + 1$ if the flip is heads, otherwise $X_{n+1} = 1/X_n$ if tails. Can we compute $\lim_{n \to…
user2566092
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chance on throwing a six with 6 dice

The chance to throw a 6 with one die is 1/6 And 6 times 1/6 = 1 So, if I throw with 6 dice, the chance to throw at least 1 six should be 1. But when I throw 6 dice, I sometimes don't throw any 6 at all.. How come?
Wouter Dorgelo
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expected number of coin flips given a condition

I have a fair coin, and I flip it until the following condition is met: #heads - #tails = N OR #tails - #heads = N where $N \geqslant 2$. What is the expected number of times I flip the coin?
Robz
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Probability of balls in boxes

If $12$ balls are thrown at random into $20$ boxes, what is the probability that no box will receive more than $1$ ball? So my book says the answer is: $\displaystyle \frac{20!}{8!20^{12}}$ However I am having some trouble understanding this result…
user181304
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What's the name of this stochastic process?

I heard about it sometime somewhere and want to read about it now, but I can't recall what the name is: Start with $a_1 = \ldots =a_n=1$. Choose a number between 1 and $n$ with probability $a_i/(a_1+ \ldots + a_n)$ to choose $i$. If $i_0$ is the…
user3533
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Independence of sum and difference for random vectors

Suppose I have two independent random vectors $X,Y\in \mathbb R^n$ such that each entry $x_i\sim\mathcal{N}(0,1)$ i.i.d., $y_i\sim\mathcal{N}(0,1)$ i.i.d. Define $$ S = X+Y, \qquad D = X-Y .$$ I seem to remember learning that $S$ and $D$ are…
mass
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Probability of a m-long cycle

You have a deck of numbered cards from $1$ to $n$. After shuffling the cards randomly, you put them in numbered boxes, from $1$ to $n$, i.e. $1$ card per box. So box $1$ can contain any one card from $1$ to $n$, etc. You now go to box $1$ and look…
Demian Schkolnik
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Monkeys and Typewriters

Suppose that there is a certain collected works of plays that is N symbols long in the following sense: a "symbol" is one of the 26 letters of the alphabet, a line break, period, space, or a colon; in other words there are 30 possible symbols. If "a…
Matt Calhoun
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Probability that ace of spades is at bottom of deck IF ace of hearts is NOT at top

What is the probability that the ace of spades is at the bottom of a standard deck of 52 cards given that the ace of hearts is not at the top? I asked my older brother, and he said it should be $\frac{50}{51} \cdot \frac{1}{51}$ because that's…
Jeff
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What is the difference between a field and a sigma field?

Can someone explain what is the smallest sigma field? I need to know this Thanks
Olórin
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If $X_i$ are iid, finding $E(X_1 + X_2 + \cdots + X_k \mid X_1 + X_2+ \cdots +X_n=b)$

I just wonder if anybody can help me to prove the following identity: Given a series of i.i.d. non-negative random variables $X_1, X_2, ..., X_n$, then $$E(X_1+X_2+ \cdots +X_k \mid X_1+X_2+ \cdots +X_n=b)=b \cdot \frac{k}{n} .$$
Qiang Li
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Maximizing heads/number of flips game

Flip a coin until you wish to stop. Your goal is to maximize the ratio number of heads/total number of flips. What is the expected value of this game? Additionally, how would one play this game?
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