Mathematics: Probabilistic and Statistical Reasoning Study Guide for the TSIA2

Probability

Probability is a branch of mathematics that describes the overall chances of an event occurring, even though the event may be random. It also describes the chances of two or more events occurring if the individual probabilities are known.

Probability Distribution

For a given set of events, a probability distribution assigns a probability to each individual event. If the probability is the same for each event, the distribution is uniform. For example, in a standard 52 card deck, the probability of drawing any particular card at random is \(\frac{1}{52}\), and the distribution is uniform.

Conditional and Independent Probabilities

If the outcome of event B depends on the outcome of event A, the probability of event B is said to be conditional. For example the probability of drawing an 8 from a standard 52 card deck on the second draw will be different depending on whether an 8 was drawn on the first draw.
In contrast, if the events A and B do not affect each other in any way, the probabilities are independent. If two standard dice are thrown, the outcome of one die is independent of the other and the probabilities are independent.

The Rules of Probability

There are several rules that govern probability calculations. Here are explanations of these rules.

Addition Rule

For two events A and B, the addition rule is used to determine the probability that at least one of them will occur, in other words the probability of A or B occurring. The addition rule states that you add the individual probabilities then subtract the probability that they both occur together:

the P(A or B) = P(A) + P(B) - P(A and B)

Multiplication Rule

For two events A and B, the multiplication rule is used to determine the probability that both events occur, the probability of A and B.

For conditional events, the multiplication rule states that you multiply the probability that A occurs times the probability that B occurs given that A occurs:

For independent events, the rule states that you multiply the individual probabilities:

P(A and B) = P(A) x P(B)

A Complement in Probability

For a given event A with probability P(A), the complement of that event is not A and its probability is 1 - P(A). For example, say you rolled a single cubical die and it came up a 4. The complement of that event is the event not 4. In other words, it is the event 1, 2, 3, 5, or 6.

The probability of rolling a 4 is 1/6 because there is only 1 way to roll a 4. There are five ways to roll not 4, so the probability of rolling not 4 is 5/6 . If you add the probabilities of an event and its complement, the answer is always 1.

The complement of event A is denoted A’. We can restate the sentence above as:

P(A) + P(A’) = 1