Articles of probability

How do you calculate $P(A|B)$ if you know only $P(A), P(B|\neg A)$?

I’ve been studying probability to develop a more intuitive sense of calculating probabilities as a medical practitioner. One example that came up in discussing the importance of prior probabilities was HIV testing. In the example the book The Laws of Medicine by Siddhartha Mukherjee gives, he gives the problem of calculating the probability of someone […]

How likely/unlikely is an event with probability $1$/$0$?

Probability theory says that if an event $E$ is certain to happen, then $P(E)=1$ which makes sense. Similarly, an impossible event has probability $0$. What surprised me is the fact that you can still find mathematical texts (notice that this paper comes from a renowned American university) that say the converse are also true, namely: […]

Using Stirling's formula to uniformly bound Bernoulli success probabilities

In this paper, the authors say that for any $\gamma \in [1/2,1)$, there is a positive constant $B=B(\gamma)$ such that for any $n$, $$ \sum_{n\gamma\leq k \leq n} \binom{n}{k} \leq B n^{-1/2}2^{n \cdot h(\gamma)}, $$ where $h(x)=-x\log_{2} x – (1-x)\log_{2}(1-x)$. They also say that that fact follows immediately from Stirling’s formula. My question is why […]

Probability of equal no. of red/black cards from selection – simulation vs. answers discrepancy

Following reading this thread: “Probability of drawing exactly 13 black & 13 red cards from deck of 52”, I created a simple simulation using Excel/VBA to help my son grasp the concept – he’s only 7 but wanted to know more… In this simulation, I chose 2, 4, 6…50 cards from a deck of 52 […]

Calclute the probability?

A random function $rand()$ return a integer between $1$ and $k$ with the probability $\frac{1}{k}$. After $n$ times we obtain a sequence $\{b_i\}_{i=1}^n$, where $1\leq b_i\leq k$. Set $\mathbb{M}=\{b_1\}\cup\{b_2\}\cdots \cup\{b_n\}$. I want to known the probability $\mathbb{M}\neq \{1, 2\cdots, k\}$.

Finding the mode of the negative binomial distribution

The negative binomial distribution is as follows: $f_X(k)=\binom{k-1}{n-1}p^n(1-p)^{k-n}.$ To find its mode, we want to find the $k$ with the highest probability. So we want to find $P(X=k-1)\leq P(X=k) \geq P(X=k+1).$ I’m getting stuck working with the following: If $P(X=k-1)\leq P(X=k)$ then $$1 \leq \frac{P(X=k)}{P(X=k-1)}=\frac{\binom{k-1}{n-1}p^n(1-p)^{k-n}}{\binom{k-2}{n-1}p^{n}(1-p)^{k-n-1}}.$$ First of all, I’m wondering if I’m on the right […]

What is the chance of picking y in a set of x objects, given x chances to pick at random?

Suppose you have a set of x objects. In it is an object y. Suppose you pick at random one object from this set. This set is uniformly distributed. There is a 1 / x chance of picking y. Let’s say that you pick x times from the set, then eliminating it from the set. […]

Error propagation, why use variences?

I have been reading up on error propagation and am slightly confused about something. We can the error in $c=f(a,b)$ as the: $$\sigma(c)= f_a \sigma_a+f_b \sigma _b$$ Firstly is this correct and am I correct in saying that the partial derivatives are evaluated at the mean of $a$ and $b$? Ever where I look, however, […]

Probability: best chance of picking a desired marble out of 10

following are the two questions I’ve made myself, but I need help in solving them. 1) Suppose there are 10 marbles in a box. One out of them is the desired marble, or you can say one is black others are white,etc etc. Case 1 : You pick 3 marbles altogether out of the box. […]

Statistics resources with examples for a C.S. student

I’m a computer science student and is fairly familiar with basic probability (calculating the probability of a event occurring, pmfs and pdfs) but I find it very difficult to grasp the concepts of advanced probability like principles of data reduction (sufficiency, likelihood principle, etc), point and interval estimation, Hypothesis testing, etc. I think it is […]