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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 P(A and B)

I have difficult to understand the following rule. Can anyone use a simple example to explain the rule to me? Thanks. If A and B are any two events, then $P(A \cap B) = P(B)\cdot P(A| B)=P(A)\cdot P(B|A)$, where P(B|A) is the probability of B…
math
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Proof that $\frac{S_n}{n}$ converges almost surely to $\mu$

I'm trying to show that given $(X_i)$ i.i.d., $E[X_i^2] < \infty$, $\mu := E[X_i]$ then $P\Big [ \lim_{n \rightarrow \infty} \frac{S_n}{n} = \mu \Big ] = 1$ where $ S_n := \sum_{k=1}^n X_k$. So far, I have rewritten $P\Big [ \lim_{n \rightarrow…
Rudy the Reindeer
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What's the probability that the next coin flip is heads?

I randomly chose between 2 coins. One of the coins has a 0.8 chance of heads and a 0.2 chance of tails. The other is a fair coin that has a 0.5 chance of either heads or tails. I flip this coin twice and get 2 heads. What's the probability that my…
John Hoffman
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Probability of drawing the same color marble twice in a bag of $10$ red, $10$ orange, $10$ green.

The question is worded as follows. A bag of marbles contains $10$ red marbles, $10$ orange marbles, and $10$ green marbles. You randomly take one marble out of the bag for yourself, and without replacement you then randomly take another marble out…
Matt Kelly
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Probability of getting an odd number of heads if n biased coins are tossed once.

The question is basically to find out the probability of getting odd number of heads when $ n$ biased coins,with $m^{th}$ coin having probability of throwing head equal to $\frac{1}{2m+1}$ ($m=1,2,\cdots,n$) are tossed once.The results for each coin…
Navin
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On average, how many times will I need to roll a six-sided die before I see ten ONES in total?

On average, how many times will I need to roll a six-sided die before I see ten ONES in total? I'd like to be able to extend my knowledge to answer other questions in the same form, such as, "If only 15% of fish are within the legal size limit, how…
Maezar
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True Randomness and repetition

I do not have a degree in any field of mathematics; however I would like to get an input perhaps from those who do. I argued a point with one of my children the other day that if all of the arguments against a system that could generate true random…
Sabre
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Trouble understanding conditional probability question

This is a solved example in Introduction to Probability by Tsitsiklis, page 26, example 1.11 A Class consists of 4 graduate and 12 undergraduate students is randomly divided into 4 groups of 4. What is the probability that each includes a graduate…
user115368
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Three shooters shoot a target

Three shooters shoot a target. The known probability to hit the target of the first, second and third shooter are 0.3, 0.4 and 0.5 respectively. What is the probability that the target is hit from only one shooter? I try this. The probabilities that…
vachoh
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Probability related to circle

Let we have to select point nearer to center then the circumference of a given circle. What is the probability of finding such points? I am confused that points at distance$\frac{r}{2}$must be favorable or not
Blaise Thunderstorm
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Finding probability using moment-generating functions

I'm working through Schaum's Outline of Probability, Random Variables, and Random Processes, and am stuck on a question about moment-generating functions. If anyone has the 2nd edition, it is question 4.60, part (b). The question gives the…
konfushus
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Probability against winning a raffle

"If there are 10000 raffle tickets, all of which are sold, and you purchased 20 of these tickets, what are the odds against you winning?" This was a question I got wrong on a recent test which I plan to retake (as an altered version of the…
Devster44
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Why is the probability of having 2 boys 7/15?

A couple plans on having 2 children. Given that at least one of them is a boy, the probability that both are boys is $$ \frac{P(both~boys)}{P(at~least~one~is~a~boy)} = \frac{0.25}{0.75} =\frac{1}{3} $$ Furthermore, a textbook I am reading claims…
Minden Petrofsky
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Roll 10 fair die. What is the probability that the number 1 appears exactly four times, on four consecutive rolls

I am asked: Roll $10$ fair die. What is the probability that the number $1$ appears exactly four times, on four consecutive rolls? The answer I was given is $$\frac{1}{6^{10}}\cdot7\cdot5^6$$ with a comment saying we multiply by 7 "for the first…
user225477
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Index where first half exceeds second half

Let $n$ be a positive integer, and independently randomize numbers $x_1,\dots,x_n,y_1,\dots,y_n$ from $(0,1)$ uniformly. Let $i(x)$ be the least index such that $$x_1+\dots+x_{i(x)}>x_{i(x)+1}+\dots+x_n.$$ Define $i(y)$ similarly. As $n$ grows, is…
pi66
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