In mathematics, the term "equally likely" refers to a situation where all possible outcomes have the same probability of occurring. It implies that each outcome has an equal chance of happening, making it a fair and unbiased scenario.
The concept of equally likely has been present in mathematics for centuries. It can be traced back to the early works of mathematicians like Blaise Pascal and Pierre de Fermat, who laid the foundation for probability theory in the 17th century. Since then, the notion of equally likely events has been extensively studied and applied in various fields, including statistics, game theory, and decision-making.
The concept of equally likely is typically introduced in elementary or middle school mathematics. It serves as a fundamental building block for understanding probability and lays the groundwork for more advanced concepts in later grades.
Equally likely encompasses several key knowledge points, including:
Probability: Equally likely events have equal probabilities. The probability of an event occurring is calculated by dividing the number of favorable outcomes by the total number of possible outcomes.
Sample Space: The sample space refers to the set of all possible outcomes in a given situation. In the case of equally likely events, each outcome in the sample space has an equal chance of occurring.
Counting Principles: Counting principles, such as the multiplication principle and the addition principle, are often used to determine the total number of possible outcomes and favorable outcomes in a scenario.
Equally likely events can be classified into two main types:
Discrete Equally Likely: In this type, the outcomes are distinct and separate from each other. For example, when rolling a fair six-sided die, each face has an equal probability of landing face-up.
Continuous Equally Likely: In this type, the outcomes form a continuous range. For instance, when selecting a random number between 0 and 1, any value within that range has an equal likelihood of being chosen.
Some properties associated with equally likely events include:
Mutually Exclusive: Equally likely events are mutually exclusive, meaning that they cannot occur simultaneously. If one event happens, the others become impossible.
Complementary: The sum of the probabilities of all equally likely events is equal to 1. This property ensures that one of the outcomes must occur.
To calculate the probability of equally likely events, divide the number of favorable outcomes by the total number of possible outcomes. The formula for probability is:
Probability = Number of Favorable Outcomes / Total Number of Possible Outcomes
There is no specific symbol or abbreviation exclusively used for equally likely events. However, the general probability symbol "P" is often employed to represent the probability of an equally likely event.
There are various methods to determine equally likely probabilities, including:
Counting: Counting the number of favorable outcomes and total possible outcomes using counting principles.
Simulation: Conducting experiments or simulations to observe the frequency of different outcomes and estimate their probabilities.
Example 1: What is the probability of rolling an even number on a fair six-sided die?
Solution: The favorable outcomes are 2, 4, and 6, while the total possible outcomes are 1, 2, 3, 4, 5, and 6. Therefore, the probability is 3/6 or 1/2.
Example 2: A bag contains 5 red marbles and 5 blue marbles. What is the probability of randomly selecting a red marble?
Solution: The favorable outcomes are selecting a red marble, and the total possible outcomes are selecting any marble from the bag. Hence, the probability is 5/10 or 1/2.
Example 3: A fair coin is tossed three times. What is the probability of getting exactly two heads?
Solution: The favorable outcomes are HHT, HTH, and THH, while the total possible outcomes are HHH, HHT, HTH, HTT, THH, THT, TTH, and TTT. Therefore, the probability is 3/8.
A spinner has 8 equal sections numbered from 1 to 8. What is the probability of landing on an odd number?
A deck of cards contains 52 cards, including 4 aces. What is the probability of drawing an ace?
A bag contains 3 red balls, 4 blue balls, and 5 green balls. What is the probability of randomly selecting a blue ball?
Q: What does it mean for events to be equally likely?
A: When events are equally likely, it implies that each event has the same probability of occurring.
Q: How can I determine if events are equally likely?
A: Events can be considered equally likely if they have the same number of favorable outcomes and the same total number of possible outcomes.
Q: Can equally likely events have different outcomes?
A: No, equally likely events must have the same outcomes, but they can occur in different orders or arrangements.
Q: Are equally likely events always fair?
A: Equally likely events are fair in the sense that they provide an unbiased and equal chance for each outcome to occur. However, fairness can also depend on other factors, such as the context or rules of the situation.