Articles of probability

How many turns, on average, does it take for a perfect player to win Concentration (adjusted)?

Let’s say a person with perfect memory is playing the card-matching game Concentration. He/she randomly lays out 2n cards (ie. n card pairs) sorted randomly and repeatedly takes turns consisting of flipping over two cards, one at a time (so you may look at the first card before flipping a second). When the cards match […]

Probability returning to initial state

Let $P=\begin{bmatrix}0&\frac{1}{2}&\frac{1}{2}\\\frac{1}{2}&0&\frac{1}{2}\\\frac{1}{2}&\frac{1}{2}&0\end{bmatrix}$ and $P^{(n+1)}=P^{(n)}P.$ I know that if you start in any vertex the probability of return after $n$ hops is the same $P_{11}^{(n+1)}=P_{22}^{(n+1)}=P_{33}^{(n+1)}$ then $$p_{11}^{(n+1)}=\frac{1}{2}p_{12}^{(n)}+\frac{1}{2}p_{13}^{(n)}$$ since $p_{12}^{(n)}+p_{13}^{(n)}=1$ $$p_{11}^{(n+1)}=\frac{1}{2}\left(1-\frac{1}{2}p_{12}^{(n)}\right)$$ but I do not know how to solve this recurrence relation or if is right.

Poisson process. Time between two events.

Suppose that people immigrate to a territory according to a Poisson process with a $\lambda =$ rate of 1 per day. What is the probability that the time between the tenth and eleventh exceeds two days?

Is the definition if mutual independence too strong?

Let $A_1,\ldots,A_n$ be $n$ events in a discrete probability space. Can someone give me an example such that $$P(A_1 \cap \cdots \cap A_n)=P(A_1)\cdot \ldots \cdot P(A_n)$$ holds, but such that there is a subset $A_{i_1},\ldots,A_{i_k}$of the $A_1,\ldots,A_n$, such that $$P(A_{i_1}\cap\ldots \cap A_{i_k})\neq P(A_{i_1})\cdot\ldots \cdot P(A_{i_k}) \ \ ?$$ If that weren’t such an example that […]

Cramér's Model – “The Prime Numbers and Their Distribution” – Part 2

Following a previous question (here you’ll find an introduction): The book states that using the convergence of the binomial distribution towards the Poisson distribution, it’s easy to show that $$|\{x\le\xi:\pi_S(x+\lambda \log x)-\pi_S(x)=k\}|\sim\xi\mathrm e^{-\lambda} \frac{\lambda^k}{k!}\quad(\xi\to\infty)$$ holds almost surely. I couldn’t prove this.

Probability of collecting all 4 different items while picking 1 random item from the set

a.) There are $4$ distinct items in the set. What is the probability of picking all $4$ items after picking $n\ge4$. b.) How many items do you need to pick to collect all four with a probability of at least $.9$. My answer for part a. (which I think is wrong): The sample space is […]

Techniques for evaluating probability integral

Consider the integral of a normal distribution: $$\int_a^b f(x)\,\mathrm d x=c $$ and a second integral for the expected value: $$ \int_a^b x\cdot f(x)\,\mathrm dx $$ Since you know the first integral is equal to $c$, what is a good way to evaluate the second integral to find the expected value? Integration by parts doesn’t […]

How to calculate the following conditional expectation? Is my calculation process right?

I want to calculate the conditional person’s correlation coefficient. But I don’t know how to calculate the following expressions,especially the conditional expectation of $E[XY|X>=F^{-1}_X(p),Y>=F^{-1}_Y(q)]$. Who can help me? I want use their joint CDF to express the final result. Thanks. Is my calculation process right?

Calculate probability of passing in an exam

I need to calculate the probability of passing on an exam. Each exam chooses 30 random questions out of 5500 questions. If you miss more than three questions, you fail. How can I calulate the probability of passing the exam? I am able to get the number of aproved exams, the number of failed exams, […]

Probability using bayes' theorem and without using bayes' theorem – part 2

Extension to the problem There are three boxes. First box has 2 green and 4 black balls. Second box has 1 green and 2 black balls. Third box contains 5 green and 4 black balls. probability of selecting first box is 1/3 . Probability of selecting second box is 1/6. Probability of selecting third box […]