"The big problem in probability theory, and particularly stochastic processes, is not so much how to solve well-posed problems. Anybody can do that. The hard problem is finding the right models for a real-world problem."
"What is probability physically, where does it come from?"
"Because the more times that you do something even if it's really unlikely, eventually it becomes a mathematical certainty."
"You expand probabilities by focusing; collapse them by letting go."
"Keep everything, keep it because you'll look back and think Wow."
"The idea that a monkey could write Hamlet by accident seems counterintuitive, but in fact it is mathematically provable when one considers the probabilities." - Monkeys and Typewriters
"An infinite universe undermines all probabilities because when everything happens an infinite number of times, nothing is more or less likely."
"Prior probability is high, that's how you do it. Background evidence establishes the higher prior and makes the theory easy to demonstrate."
"In a big enough sample size any event that can happen will happen, and a bigger one will happen again."
"It's like when we flip a coin, it's a 50-50 chance that we'll get a head or a tail, but if we flip it twice, we're not necessarily going to actually get a head and a tail, we could get two tails."
"Each flip has a 50-50 chance for either result independent of the previous flips."
"We're talking about probability, not about possible and impossible."
"The probability that the server is idle is 1 - (lambda / mu)."
"On a direct intuitive level, the higher a priori probability assigned to a sequence with a short description corresponds to one possible interpretation of Occam's razor." - Ray Solomonoff
"The sum of probabilities for mutually exclusive outcomes would be one."
"Probability of A intersection B is equal to probability of A times probability of B."
"If a random variable X has a cumulative density function F of X, then the variable F inverse of U, where U is a random uniform variable between 0 and 1, also has CDF F of X."
"We interpret this volume as the probability of a random vector following C to fall in this little cube."
"The sum of all those probabilities must equal 1."
"Probability is simply trying to quantify the uncertainty in a random process."
"The word 'and' basically means multiply, and the word 'or' in probability means add."
"That's really the meaning of convergence in probability."
"Mean square convergence implies convergence in probability."
"Via Bayes rule, we can infer information about situations that we didn't initially think about."
"The trick here is that we clearly defined the uncertainty that we have, and we can infer the rest by just applying probability theory."
"Every probability has to be between 0 and 1 inclusive."
"We're going to use probability theory to perform our predictions."
"Probability theory is fascinating in that on the one hand it's described by these two incredibly simple rules, and yet that can lead to some extraordinary and complex ramifications."
"Probabilities used in this manner is the only consistent way of doing what we call plausible reasoning."
"Pascal's triangle forms the basis of much of modern probability theory."
"The probability of A or B is the probability of A plus the probability of B minus the probability of A and B."
"Compound probabilities are basically when we have more than one thing that we're looking at at the same time."
"If the outcomes are equally likely, then the outcomes in the sample space diagram are equally likely."
"Mutually exclusive events mean that only one of them has the possibility of occurring at any time."
"The Law of Large Numbers and Central Limit Theorem are possibly the two most important theorems in probability."
"Mutually exclusive events are events which cannot occur simultaneously."
"Randomization allows you to use the probability theory."
"An event is independent of the other if the probability of their intersection is the product of their individual probabilities."
"The probability that a randomly selected node will have a certain degree."
"When you increase P, the probability that two nodes are connected, you increase on average the average node degree in the graph."
"The sum of all the probabilities is equal to 1."
"The probability distribution function of a random variable means the values taken by the random variable and their associated probabilities."
"The variance of x is equal to the sum of x minus the mean, squared, multiplied by the probability of x."
"A collection of events is mutually independent if any knowledge about any of the rest of the events, happening or not, does not influence the event you're looking at for each of those events."
"If all events are equally likely, the probability of event A is the number of occurrences that are in favor of A divided by your total outcomes."
"If you gave me the task of inventing a generalization of probability theory where the probabilities could be negative, you would invent quantum mechanics."
"The multiplication rule says if you want to find the probability of A and B, you just take the probability of A times the probability of B."