The Mathematics of Maybe: Introduction to ProbabilityClass 9 Mathematics Important Points

Probability is a measurement of the likelihood of an event, expressed on a scale from 0 to 1. For any event E, its probability $P(E)$ must satisfy the condition $0 \leq P(E) \leq 1$.

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

Theoretical probability is used when all outcomes are equally likely. It is calculated with the formula $P(\text{Event}) = \frac{\text{Number of favourable outcomes}}{\text{Total number of possible outcomes}}$.

Experimental probability, or relative frequency, is based on data from trials. It is calculated as $P(\text{Event}) = \frac{\text{Number of times the event occurred}}{\text{Total number of trials}}$.

The sample space, denoted by S, is the set of all possible outcomes of a random experiment. Each individual outcome is an element of the sample space.

An event is a specific result or a collection of results from a random experiment. An event is always a subset of the sample space.

Outcomes are considered equally likely if there is no reason to believe one will occur more often than another. This is the foundation for calculating theoretical probability, such as with a fair coin or die.

This principle states that as the number of trials in an experiment increases, the experimental probability of an event will tend to get closer to its theoretical probability.

This is the mistaken belief that past outcomes of independent random events can influence future ones. Each flip of a fair coin or roll of a fair die is an independent event with no memory of past results.

A tree diagram is a visual tool used to list all possible outcomes in an experiment that involves multiple steps. Each complete path from the start to an end point represents one outcome in the sample space.

For a single toss of a fair coin, the sample space is S = {Heads, Tails}. The probability of getting heads is $P(\text{Heads}) = \frac{1}{2}$ and the probability of getting tails is $P(\text{Tails}) = \frac{1}{2}$.

For a single roll of a fair 6-sided die, the sample space is S = {1, 2, 3, 4, 5, 6}. The probability of rolling any specific number is $\frac{1}{6}$.

When rolling a fair 6-sided die, the favorable outcomes for an even number are {2, 4, 6}. The probability is $P(\text{even number}) = \frac{3}{6} = \frac{1}{2}$.

When two coins are tossed simultaneously, the sample space consists of four possible outcomes: S = {HH, HT, TH, TT}, where H is Heads and T is Tails.

When tossing two coins, the event 'at least one head' includes the outcomes {HH, HT, TH}. The probability is $P(\text{at least one head}) = \frac{3}{4}$.