In this section, we will look at
Probability: the fundamentals of probability, conditional probability, Venn Diagrams
Counting Problems: combinations, permutations and binomial coefficients.
Random Variables: Introduction to random variables, sampling.
The conditional probability of an event is the probability that an event A occurs given that another event B has already occurred. This type of probability is calculated by restricting the sample space that we’re working with to only the set B.
The formula for conditional probability can be rewritten using some basic algebra. Instead of the formula:
\[P(A | B) = \frac{P(A \cap B) }{P( B )} \]
The multiplication rule is a result used to determine the probability that two events, \(A\) and \(B\), both occur. The multiplication rule follows from the definition of conditional probability.\
The result is often written as follows, using set notation: \[ P(A|B)\times P(B) = P(B|A)\times P(A) \qquad \left( = P(A \cap B) \right) \]
Bayes’ Theorem is a result that allows new information to be used to update the conditional probability of an event.
Recall the definition of conditional probability: \[ P(A|B) = \frac{P(A \cap B)}{P(B)} \]
Using the multiplication rule, gives Bayes’ Theorem in its simplest form:
\[ P(A|B) = \frac{P(B|A)\times P(A)}{P(B)} \]
The general form of Bayes’ theorem is \[ P(A|B) = \frac{P(A \mbox{ and }B)}{P(B)} \]
Equivalently Bayes’ Theorem can be expressed as \[\begin{equation*} P(B|A)=\frac{P\left(A|B\right) \times P(B) }{P\left( A\right) }. \end{equation*}\]Conditional Probability
**{Basics of Probability} %====================================%
***{Random Experiment} A random experiment is one whose outime is determined by chance.
***{Sample Space} In a probabilistic experiment, the sample space is the set of all possible outcomes of the experiment. Suppose the probabilistic experiment is the toss of a dice. The six numbers that can appear face up, from 1 to 6, are the 6 possible outcomes of the experiment. Hence, the sample space is: \[S={1,2,3,4,5,6}\]
***{Sample Points} In a probabilistic experiment, a sample point is one of the possible outcomes of the experiment. The set of all sample points is called sample space.
**{Events} An event A* is merely collection of outcomes, or in other words, a subset of the sample space.
***{Sample Spaces and Events}
A card is drawn at random from a pack of cards. Let A be the event that the card is red and B be the event that the card is king.
Does the colour of card affect its probability of being a king?
Suppose that A and B are two independent events with P(A), the probability that A occurs, equal to 0.4, and P(B), the probability that B occurs, equal to 0.5. You may assume that A and B are independent events.
Suppose that \(A\) and \(B\) are events from a sample space such that \(P(A) = 0.65\) and \(P(B) =0.45\) and \(P(A\cap B) = 0.32\)
Solution
Consider the random variable \(\LARGE{X}\) taking the value \(\LARGE{X = 1}\) if a randomly selected person is a smoker, or \(\LARGE{X = 0}\) otherwise.
The joint distribution of \(\LARGE{X}\) and \(\LARGE{Y}\) is given by the joint probability function in the following table.
| Y= 0 | Y= 1 | Y= 2 | ||
|---|---|---|---|---|
| X = 0 | 0.2 | 0.3 | 0.25 | |
| X = 1 | 0.1 | 0.1 | 0.05 | |
The random variable \(\LARGE{R = (3 - Y)^2(X + 1) }\) is used as a risk index for a particular heart disease.
Exercises