Theoretical Probability Formula

The theoretical or classical probability of an event E is the ratio of favorable outcomes to total outcomes. The formula is $P(E) = \frac{\text{Number of outcomes favourable to E}}{\text{Total number of possible outcomes}}$ .

Range of Probability

The probability of any event E is a number between 0 and 1, inclusive. This is represented as $0 \leq P(E) \leq 1$ .

Sure and Impossible Events

An event that is certain to happen is a sure event and its probability is 1. An event that cannot happen is an impossible event and its probability is 0.

Complementary Events

For any event E, the event 'not E' is its complement, denoted by $\bar{E}$ . The sum of their probabilities is always 1, so $P(E) + P(\bar{E}) = 1$ . This implies $P(\bar{E}) = 1 - P(E)$ .

Elementary Events

An event having only one outcome of an experiment is called an elementary event. The sum of the probabilities of all the elementary events of an experiment is 1.

Equally Likely Outcomes

The definition of theoretical probability assumes that all possible outcomes of an experiment are equally likely. This means each outcome has the same chance of occurring, like in a fair coin toss or a fair die roll.

Standard Deck of Playing Cards

A standard deck has 52 cards, divided into 4 suits (Spades, Hearts, Diamonds, Clubs) of 13 cards each. There are 26 red cards (Hearts, Diamonds) and 26 black cards (Spades, Clubs). There are 12 face cards (King, Queen, Jack).

Rolling a Single Die

When a single fair die is thrown once, there are 6 equally likely outcomes: ${1, 2, 3, 4, 5, 6}$ . The probability of any single number appearing is $\frac{1}{6}$ .

Rolling Two Dice

When two dice are thrown simultaneously, the total number of possible outcomes is $6 \times 6 = 36$ . The outcomes are represented as ordered pairs, for example, $(1, 4)$ is different from $(4, 1)$ .

Tossing Coins

Tossing one coin gives 2 outcomes (H, T). Tossing two coins simultaneously gives 4 outcomes: (HH, HT, TH, TT). Tossing three coins gives 8 outcomes: (HHH, HHT, HTH, THH, HTT, THT, TTH, TTT).

Probability Cannot Be Negative

A value less than 0, such as -1.5, cannot be the probability of an event. Similarly, a value greater than 1, such as $110\%$ , cannot be a probability.

Event 'At Least One'

To find the probability of 'at least one' of something occurring, it is often easier to calculate the probability of its complement, 'none', and subtract from 1. For example, $P(\text{at least one head}) = 1 - P(\text{no heads})$ .

Key Points