Probability¶

Experiments, Sample Spaces and Events¶

For any experiment, outcome is uncertain.
“Sample Space - The set of all possible outcomes for an experiment.”
A set of possible outcomes i.e. a subset of points in a sample space in a probabilistic experiment is called an event.

“Mutually Exclusive Events - A set of events are mutually exclusive if the occurrence of one event precludes the occurrence of any other event.”
If the events are mutually exclusive then, \(P(E_1\) or \(E_2) = P(E_1) + P(E_2)\)
If the events are not mutually exclusive then, \(P(E_1\) or \(E) = P(E_1) + P(E_2) - P(E_1\) and \(E_2)\).

“Two events are complementary events if they have no points in common and together include all points in an experiment’s sample space.”
The complement of event \(E\) is \(\bar{E}\).
\(E\) and \(\bar{E}\) are mutually exclusive events.
\(P(E\) or \(\bar{E}) = 1 = P(E) + P(\bar{E})\).

“Given two events \(A\) and \(B\) the conditional probability of event \(A\) given that event \(B\) has occurred is written as \(P(A|B)\). Intuitively, once we know that event \(B\) has occurred, this is the chance that event \(A\) will occur.”
\(P(A|B) = \frac{P(A and B)}{P(B)}\)
“Given two events A and B we define the joint probability of A and B to be the probability that events A and B both occur.”

“Independent Events - Two events are independent if the occurrence of one of the events does not change our estimate of the probability of the other event. In short, events A and B are independent if and only if P(A|B) = P(A).”
For two independent events \(E_1\) and \(E_2\), \(P(E_1\) and \(E_2) = P(E_1) \times P(E_2)\)

“Random Variable - A function that associates a numerical value with every possible outcome of an experiment.”
Random variables can be discrete or continuous.
“Expected Value of a Random Variable - The average value you would expect to see of a random variable if you perform an experiment many times.”
“The expected value of a discrete random variable is found simply by multiplying each value of the random variable by its probability and then adding up the products.”
For discrete random variables, \(E(X) = \sum_{i=1}^{n} p_i x_i\).
“The expected value of a discrete random variable tells you, on average, what value that variable has.”
“Variance of a Random Variable - A measure of a random variable’s spread about its mean defined as the average squared deviation of a random variable from its mean. \(\sigma^2(X) = \sum_{i=1}^{n} p_i (x_i - E(X))^2\).”
The standard deviation of a random variable \(\sigma(X)\) is simply the square root of the variance.

“Continuous Random Variable - A continuous random variable is used to describe an uncertain quantity (such as height, weight, or return on a stock) which can assume an infinite number of values and is defined over an interval or intervals of values.”
“Probability density function (PDF) - A continuous random variable’s pdf tells us the relative likelihoods of the random variable’s possible values. The area under a pdf between a and b is the probability that the continuous random variable assumes a value between a and b.”
A PDF has the following properties:
- It is always nonnegative.
- The height of a PDF for a value x of a continuous random variable represents the relative likelihood that the random variable assumes a value near x.
- The total area under the PDF must equal 1.
- The probability of an event involving a continuous random variable corresponds to the area under the PDF. The total probability of all possible outcomes must equal 1, in line with the total area of 1 under the PDF.

“Normal Random Variable - A continuous random variable that describes many quantities such as height, weight, or monthly sales of a product. A normal random variable is specified by its mean and standard deviation. The Excel function =NORMDIST(x, mean, standard deviation, True) gives the probability that a normal random variable with a given mean and standard deviation is ≤ x.”
“The normal random variable, often called the bell curve, has the following PDF: \(f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{- \frac{(x - \mu)^2}{2 \sigma^2}}\). Where, \(e=2.7182\).”
The normal random variable has several important properties:
- A normal random variable assumes a mean value μ.
- A normal random variable has a variance of \(\sigma^2\) and a standard deviation of \(\sigma\).
- There is a 68% chance that a normal random variable assumes a value within σ of the mean, a 95% chance that it assumes a value within 2σ of the mean, and a 99.7% chance that it assumes a value within 3σ of the mean.
- The normal PDF is symmetric about the mean: It looks the same to the left of the mean as it does to the right.