4.3-Chance and Probability
4.3-Chance and Probability Important Formulae
You are currently studying
Grade 8 → Math → Data Handling → 4.3-Chance and Probability
4.3 - Chance and Probability
- Chance refers to the likelihood of an event occurring, ranging from 0 (impossible) to 1 (certain).
- Probability is a measure of the chance of an event happening.
- The probability of an event $P(E)$ is given by the formula: $P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$
- For a fair coin toss, the probability of heads or tails is $P(\text{Head}) = \frac{1}{2}$
- The sum of the probabilities of all possible outcomes in an experiment is always 1.
- Sample space: The set of all possible outcomes of an experiment.
4.3 - Chance and Probability
In this section, we will learn about the concepts of chance and probability, which deal with the likelihood of an event occurring. These concepts are essential in understanding and analyzing various outcomes in everyday situations, such as tossing a coin or rolling a die.
What is Chance?
Chance refers to the likelihood or possibility of an event happening. It is often expressed as a ratio or a percentage. For example, if you toss a coin, the chance of getting heads or tails is 50%, or 1 out of 2. The higher the chance, the more likely the event will occur.
What is Probability?
Probability is a numerical value that represents the likelihood of an event. It is a measure of how likely an event is to occur and is always a number between 0 and 1, where 0 indicates that the event will not happen, and 1 means it is certain to happen. The probability of an event is calculated using the following formula:
Probability $P(E)$ of an event $E$ is given by:
$P(E) = \dfrac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$
Types of Events
Events can be classified into different types based on their nature:
- Simple Event: An event that consists of a single outcome. For example, rolling a 3 on a die.
- Compound Event: An event that consists of more than one outcome. For example, rolling an even number on a die.
- Impossible Event: An event that cannot happen. For example, rolling a 7 on a standard die.
- Certain Event: An event that is certain to happen. For example, drawing a card from a deck and getting a card that is either red or black.
Experimental Probability
Experimental probability is the probability of an event occurring based on experimental or observed data. It is calculated by performing an experiment and recording the outcomes. The formula for experimental probability is:
$P(E) = \dfrac{\text{Number of times event E occurred}}{\text{Total number of trials}}$
Example of Experimental Probability
If a die is rolled 100 times and the number 4 comes up 20 times, the experimental probability of rolling a 4 is:
$P(\text{4}) = \dfrac{20}{100} = 0.2$
Theoretical Probability
Theoretical probability, on the other hand, is based on reasoning or logic, and it is determined without conducting an actual experiment. It is based on the assumption that all outcomes are equally likely. For example, when rolling a fair six-sided die, the theoretical probability of rolling a number between 1 and 6 is:
$P(\text{any number from 1 to 6}) = \dfrac{1}{6}$
Probability of Multiple Events
When dealing with multiple events, we can calculate the probability of their occurrence using different methods:
- Independent Events: If the occurrence of one event does not affect the occurrence of the other, the events are said to be independent. For example, tossing two coins. The probability of both coins landing heads is:
- Dependent Events: If the occurrence of one event affects the occurrence of the other, the events are dependent. For example, drawing cards from a deck without replacement. The probability of drawing a red card and then a black card is:
$P(\text{Heads on Coin 1 and Heads on Coin 2}) = P(\text{Heads on Coin 1}) \times P(\text{Heads on Coin 2}) = \dfrac{1}{2} \times \dfrac{1}{2} = \dfrac{1}{4}$
$P(\text{Red then Black}) = P(\text{Red}) \times P(\text{Black after Red}) = \dfrac{26}{52} \times \dfrac{26}{51}$
Conclusion
Understanding the concepts of chance and probability helps us make better predictions and decisions based on the likelihood of events. It is widely used in various fields, such as gaming, sports, statistics, and even in daily life.