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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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Prove that a martingale bounded in $L_2$ converges almost surely

Hi as the topic states prove that a martingale bounded in $L_2$ converges almost surely.
Rachael HuiChee
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Find an example where the random variables $X_1,X_2,X_3$ are pairwise independent, but not all together.

Find an example where the random variables $X_1,X_2,X_3$ are pairwise independent, but not all together. I can't really understand how I am to do so. How is it done? There cannot be any multiplication. Is it a sum? I searched for it but didn't…
Meitar
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Coffee Shop Meeting

$A$ and $B$ decide to meet at a cafe between $5$ p.m. and $6$ p.m. They agree that the person who arrives first at the cafe would wait for exactly $15$ minutes for the other. If each of them arrives at a random time between $5$ p.m. and $6$ p.m.,…
riz
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Dice outcomes probability

Two dice are rolled. What is the probability that the sum of the numbers on the dice is at least 10 Let $Z$ denote the set of successful outcomes: $Z=\{(4,6),(6,4),(5,5),(5,6),(6,5),(6,6)\}\\ \text{Sample space: } S=6^2$ So the answer gives the…
εν οίδα ότι ουδέν οίδα
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How to calculate the probability of an event when you don't know the initial probabilities?

Say there are two coins: coin A is fair, but coin B always comes up heads. A friend then flips one of the two coins, and you observe that the coin came up heads. How do you calculate the probability that this was coin A, when you don't know if your…
user541686
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Deriving the MSE [mean squared error]

The above image is from wikipedia. I'm having troubles with the third line to the 4th: question 1): How did they get from $2E[(\hat{\theta} - E(\hat{\theta})(E(\hat{\theta}) - \theta)]$ to $2(E(\hat{\theta}) - \theta)E(\hat{\theta} -…
statsguy23
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Proving negative binomial distribution has a valid pmf

I'm trying to prove $$\sum_{k=r}^\infty \binom{k-1}{r-1}p^rq^{k-r}= 1$$ I'm given a hint to use $(a+b)^m = \sum_{k=0}^\infty \binom{m}{k} a^kb^{m-k}$. I choose to let $a = -q$, $b = 1$, $m = -r$ and I get $$(1-q)^{-r} = \sum_{k=0}^\infty…
DH.
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Normal Distribution and Conditional Probability

Suppose that exam scores were distributed normally. Let the mean be 80 and standard deviation be 8. If it is known that a student's score is greater than 75, what is the probability that his score is greater than 90? I am a bit confused by the…
user26139
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Conditional probability and independent events.

In a test, an examinee either guesses or copies or knows the answer to a multiple-choice question with four choices, only one answer being correct. The probability that he makes a guess is $\frac{1}{3}$ and the probability that he copies the answer…
user24047
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Choosing 10 balls out of a box of 90 balls

A box contains $ 30$ red balls, $30$ white balls and $30$ blue balls. If $10$ balls are selected at random without replacement, what is the probability that at least one color will be missing from the selection? The answer is: Solution: Let $A_1,…
Dkova
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Chess Probability 8 rooks

You have 8 rooks. What is the probability of placing all 8 rooks on an 8 by 8 chess board with out one being able to hit each other? But there's a catch of course..one of the spaces is unavailable to be used. That being said, there could potentially…
tim
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probability of getting 50 heads from tossing a coin 100 times

folks, i am new to this forum and not a math expert. so please bear with me if am asking silly questions. The question is "probability of getting 50 heads from tossing a coin 100 times". So the answer for this is, I guess, ${100 \choose 50} (2…
Rajan
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Using CLT to calculate probability question

Suppose $53$ percent of the population prefer red socks to green. If $100$ random people are asked, what is the probability that most (at least $50$) will say prefer GREEN: So I set \begin{align*} N & = 49.5\\ n & = 100\\ p & =…
Andy
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How do you calculate the sum of combinations of 1000 dice rolls?

For two dice rolls we can calculate the number of combinations for each summed total: Rolling a 2: one chance (1&1) Rolling a 3: two chances (2&1)(1&2) Rolling a 4: three chances (3&1)(1&3)(2&2) Rolling a 5: four chances…
Adam
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Birthday problem with exactly 3 people

I have been doing a bunch of birthday problem questions, however this one has thrown a mental block my way. The questions is: What is the probability that in a room of 10 people exactly 3 people will have the same birth month as each other, while…
Tom
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