The Covid-19 pandemic had a devastating impact on many industries, causing mass job losses. However, amidst the chaos, there were some businesses that emerged unscathed and even thrived during the pandemic. These businesses can be classified as independent, meaning they were not affected by the pandemic in any way.
In this article, we will delve into the concept of independent events, the calculation formulas used to determine their probability, and provide real-life examples of their application. Additionally, we will discuss how Venn diagrams can visually represent independent events.
An event is considered independent when its occurrence does not influence the probability of another event happening. This means that two events can happen simultaneously without one affecting the other's outcome. For instance, flipping a coin and landing on heads has no impact on the likelihood of landing on tails on the next flip. Similarly, buying a car and your sister getting accepted into university are independent events, as one does not affect the other.
Other examples of independent events include:
However, it may not always be easy to determine whether two events are independent or not. To establish independence, consider if the events can occur in any order and if one event's outcome has an impact on the other.
The formula for calculating the probability of an event occurring is:
Where:
To determine the probability of two independent events happening simultaneously, we multiply the individual probabilities of each event. The formula for this is:
Where:
For example, if we have two independent events A and B, where P(A) = 0.7 and P(B) = 0.5, the probability of both events occurring simultaneously is calculated as follows:
Therefore, the likelihood of both events happening is 0.35.
This formula can also determine if two events are genuinely independent. If the probability of the intersection is equal to the product of the individual probabilities, then the events are independent. Otherwise, they are not.
Venn diagrams are a useful tool for visualizing independent events. Using the formula for calculating the probability of two independent events happening simultaneously, we can represent this on a Venn diagram.
For example, if we have two events A and B, the probability of their intersection would be:
Let's put the formulas we have discussed into practice with some examples.
Example 1: Rolling a Die
Let's consider two independent events A and B involving rolling a die. Event A is rolling an even number, and event B is rolling a multiple of 2. What is the likelihood of both events happening simultaneously?
Solution:
Event A: Rolling an even number.
Event B: Rolling a multiple of 2.
Both events are independent. A die has six sides, with the possible numbers being 1, 2, 3, 4, 5, and 6. To find the probability of both events occurring simultaneously, we use the formula:
Therefore, the likelihood of both events happening is 1/3.
Example 2: Calculating Independent Events Probability
If events A, B, C and D are independent, and P(A) = 0.5, P(B) = 0.4, P(C) = 0.3, and P(D) = 0.2, what is P(A∩B∩C∩D)?
Solution:
Substituting the probabilities into the formula, we get:
Therefore, the likelihood of all four events happening simultaneously is 0.024.
When randomly selecting two students, what is the likelihood of both having a passion for mathematics? And what is the probability of one student enjoying math while the other does not?
In order to answer the first question, we must find the probability of two independent events occurring at the same time - the chance of both students liking math.
Let's define these events as A and B, and use the formula P(A ∩ B) = P(A) x P(B). Keep in mind that we are working with percentages, so we will divide the final result by 100.
So, what is the probability of both students liking math?
This means that the probability of both students liking math is 0.3 or 30%.
The second question asks for the possibility of one student liking math while the other does not. These are two separate independent events, and to find their intersection, we use the same formula as before.
The probability of the first student liking math is P(C) = 50% = 0.5, and the probability of the second student not liking math is P(D) = 40% = 0.4.
So, what is the probability of one student liking math and the other not?
The probability of one student liking math while the other does not is 0.2 or 20%.
We have learned how to calculate the probability for independent events, but are the events truly independent?
To determine if events C and D are independent, we use the formula P(C ∩ D) = P(C) x P(D). If the outcome differs from our expectations, then the events are not independent.
Let's plug in the given values and see what we get:
The question states that the intersection should be 0.60, but our calculation yields 0.27. This indicates that events C and D are not independent.
Let's take a look at some instances of independent events.
John wins the lottery and gets a new job. These two events can occur separately from each other, meaning that John can get a job without winning the lottery, or vice versa.
Jessica goes to college and gets married. Similarly, these events can happen independently of each other.
David wins a race and earns an engineering degree. Both events are not dependent on each other and can occur separately.
To sum it up, independent event probability refers to occurrences that do not affect the likelihood of another event taking place. The formula for calculating the probability of two independent events happening simultaneously is P(A ∩ B) = P(A) x P(B). To determine if events are independent, we can use the formula P(A ∩ B) = P(A) x P(B). If the result is equal to the product of the individual probabilities, then the events are independent. Examples of independent events include winning the lottery and getting a new job, going to college and getting married, and winning a race and earning an engineering degree.
Independence in probability refers to the occurrence of one event having no impact on the likelihood of another event happening.
To calculate probability for independent events, use the formula P(A ∩ B) = P(A) x P(B).
When trying to calculate the probability of an event happening on its own, it's important to consider both the number of ways the event can occur and the total number of possible outcomes. This information can help determine if the event is independent or not.
Independence in probability refers to events that can happen in any order and where one event does not influence the outcome of the other event. This is an important concept to understand when calculating the probability of an event occurring.
To determine if two events are independent, you can use the formula P(A ∩ B) = P(A) x P(B). This formula takes into account the probability of an event occurring on its own as well as the probability of both events occurring together. If the result is equal to the probability of both events occurring separately, then the events are independent. If the result is different, then the events are not independent.
Events can be considered independent when they meet the following criteria:
By understanding the concept of independence in probability and using the appropriate formula, you can accurately determine if events are independent and calculate their probability. This can be useful in various fields such as statistics, finance, and gambling.
