Probability Concepts
Trials and Outcomes
In probability theory, a trial refers to a single execution of an experiment or a random process. Each trial results in an outcome, which is a specific result of that trial.
ExampleFlipping a coin is a trial, and the outcome could be either heads or tails. Rolling a die is another trial, with possible outcomes being any number from 1 to 6.
Equally Likely Outcomes
Outcomes are considered equally likely if they have the same probability of occurring. This concept is fundamental in many probability calculations.
NoteIn a fair die, each number has an equal 1/6 probability of being rolled, making all outcomes equally likely.
Relative Frequency
Relative frequency is the ratio of the number of times an event occurs to the total number of trials conducted. It's often used as an experimental approximation of probability.
$$ \text{Relative Frequency} = \frac{\text{Number of occurrences of the event}}{\text{Total number of trials}} $$
ExampleIf you flip a coin 100 times and get 53 heads, the relative frequency of heads is 53/100 = 0.53.
Sample Space (U)
The sample space, denoted as U, is the set of all possible outcomes of an experiment.
ExampleFor a single coin flip: U = {Heads, Tails} For rolling a six-sided die: U = {1, 2, 3, 4, 5, 6}
Sample spaces can be represented in various ways:
- Lists: U = {1, 2, 3, 4, 5, 6}
- Tables:
- Tree diagrams:
Representing sample spaces visually can help in understanding complex probability scenarios.
Events
An event is a subset of the sample space. It's a collection of outcomes that we're interested in.
ExampleWhen rolling a die, the event "rolling an even number" is {2, 4, 6}.
Probability Calculations
Basic Probability Formula
The probability of an event A is given by:
$$ P(A) = \frac{n(A)}{n(U)} $$
Where $n(A)$ is the number of favorable outcomes (outcomes in event A) and $n(U)$ is the total number of possible outcomes (size of the sample space).
ExampleIn a standard deck of 52 cards, the probability of drawing a heart is: $P(\text{heart}) = \frac{13}{52} = \frac{1}{4} = 0.25$
Complementary Events
For any event A, its complement A' (also written as "not A" or $\overline{A}$) is the event that A does not occur. A key property is:
$$ P(A) + P(A') = 1 $$
ExampleIf the probability of rain tomorrow is 0.3, then the probability of no rain is: $P(\text{no rain}) = 1 - P(\text{rain}) = 1 - 0.3 = 0.7$
Common MistakeStudents often forget that probabilities must sum to 1, leading to errors in complementary event calculations.
Expected Number of Occurrences
The expected number of occurrences is calculated by multiplying the probability of an event by the number of trials.
$$ E = P(A) \times n $$
Where $E$ is the expected number, $P(A)$ is the probability of event A, and $n$ is the number of trials.
ExampleIf the probability of a student being absent is 0.1 in a class of 128 students, the expected number of absences is: $E = 0.1 \times 128 = 12.8$
NoteThe expected number can be a decimal, even though in reality, you can't have a fractional number of occurrences.
Experimental vs Theoretical Probability
Theoretical probability is a numerical value that is calculated using concepts in probability theory. Experimental probability, on the other hand, is based on the results of actual experiments or observations.
ExampleTheoretical probability of getting heads on a fair coin: 0.5 Experimental probability after 100 flips resulting in 53 heads: 0.53
As the number of trials increases, the experimental probability tends to approach the theoretical probability. This concept is known as the Law of Large Numbers.
