Equally likely outcomes

4.3.2 Equally likely outcomes

Description

Quick Overview

The section discusses the concept of equally likely outcomes in probability, where each outcome in a random experiment has the same likelihood of occurring.

Standard

This section explores how equally likely outcomes arise in random experiments, illustrated with examples such as tossing a coin or rolling a die. It highlights how the probability of specific outcomes can be calculated based on their equal likelihood.

Detailed

Equally Likely Outcomes

In probability, an experiment can yield outcomes that are said to be equally likely when each outcome has an identical chance of occurring. For example, when tossing a coin, the results can be either Head or Tail, each with a probability of

\[ P(Head) = P(Tail) = \frac{1}{2} \]

Similarly, when rolling a fair die, each number from 1 to 6 has the same probability:

\[ P(n) = \frac{1}{6}, \text{ for } n = 1, 2, 3, 4, 5, 6 \]

As more repetitions of an experiment are performed, the results tend to stabilize around their expected probabilities. The section illustrates how conducting a larger number of tosses results in counts of heads and tails that tend to converge, indicating that these outcomes are equally likely. The concept is foundational in linking the practical outcomes of random experiments to the theoretical framework of probability.

Key Concepts

  • Random Experiment: An experiment with uncertain outcome.

  • Equally Likely Outcomes: Outcomes that have an equal chance of occurring.

  • Probability: A numerical expression of how likely an outcome is.

  • Event: A specific result or a set of results from an experiment.

Memory Aids

🎵 Rhymes Time

  • When you toss a coin, don't you see? Heads or tails, that's the key!

📖 Fascinating Stories

  • Imagine a fair game where coins and dice roll, each side has a fair chance, that's the goal!

🧠 Other Memory Gems

  • For outcomes, use the acronym 'ELO', which stands for 'Equally Likely Outcomes'.

🎯 Super Acronyms

Use the acronym 'P.O.E.' for Probability Outcomes Equally.

Examples

  • Tossing a coin yields two outcomes: heads or tails, each with a probability of 1/2.

  • Rolling a die has six equally likely outcomes: 1, 2, 3, 4, 5, and 6.

Glossary of Terms

  • Term: Random Experiment

    Definition:

    An experiment whose outcome cannot be predicted with certainty.

  • Term: Equally Likely Outcomes

    Definition:

    Outcomes of an experiment that have the same probability of occurring.

  • Term: Event

    Definition:

    A specific outcome or a collection of outcomes from an experiment.

  • Term: Probability

    Definition:

    A measure of the likelihood of a particular outcome occurring, ranging from 0 to 1.