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 […]

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: […]

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 […]

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 […]

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 $\mathbb{M}\neq \{1, 2\cdots, k\}$.

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 […]

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. […]

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, […]

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. […]

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 […]

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