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Independent events
Independent events in probability, independent events are events in which the outcome of one event does not affect the probability of the outcome of another event. These events are said to be independent if the probability of one event occurring is not affected by whether the other event has occurred or not. For example, if a coin is flipped and a die is rolled, the outcome of the coin flip does not affect the outcome of the die roll, and vice versa. The probability of both independent events happening together is found by multiplying the probability of each event happening individually. For example, the probability of getting tails on a coin flip and a 6 on a die roll is (1/2) x (1/6) = 1/12.
Rule for independent events
The rule for independent events states that if two events A and B are independent, meaning that the outcome of one event does not affect the outcome of the other event, then the probability of both events happening is the product of their individual probabilities. This rule is often written as:
P(A and B) = P(A) * P(B)
This formula expresses the idea that the probability of both events happening is the product of the probability of each event happening independently. For example, if the probability of event A happening is 0.3 and the probability of event B happening is 0.4, then the probability of both events happening is 0.3 * 0.4 = 0.12
It’s important to note that independence is a property of the events, not the outcomes. For example, if you roll two dice, the outcome of the first roll does not affect the outcome of the second roll, so the events of rolling a 6 on the first die and rolling a 6 on the second die are independent, the outcome of getting a 6 on the first die and the outcome of getting a 6 on the second die are not independent.
Mutually Exclusive Events
Mutually exclusive events are events in probability theory that cannot happen at the same time. This means that if one event occurs, the other event cannot occur. For example, flipping a coin and getting heads is mutually exclusive with flipping the same coin and getting tails.
In mathematical terms, mutually exclusive events are events with empty intersection, meaning that the probability of their intersection is 0. Therefore, the probability of either event happening is the sum of the individual probabilities of each event happening.
For example, if you have two events A and B that are mutually exclusive, the probability of either A or B happening is: P(A or B) = P(A) + P(B)
It’s also important to note that not all events are mutually exclusive. For example, rolling a die and getting a number less than 4, and rolling a die and getting an even number are not mutually exclusive, because it’s possible to roll a die and get a 2, which satisfies both conditions at the same time.
It is also important to point out that mutually exclusive events are a way of defining the complement of an event, if we have a set of mutually exclusive events E1, E2, E3, …, En and their union is the sample space, then P(Ei) + P(Ei’) =1
Independent Events Vs Mutually Exclusive Events
| Independent Events | Mutually exclusive events |
| They cannot be specified based on the outcome of a maiden trial. | They are independent of trials |
| Can have common outcomes | Can never have common outcomes |
| If A and B are two independent events, thenP(A ∩ B) = P(B) .P(A) | If A and B are two mutually exclusive events, thenP(A ∩ B) = 0 |
Probability of two independent events
The probability of two independent events A and B happening is the product of their individual probabilities. This is known as the multiplication rule for independent events. The formula is:
P(A and B) = P(A) * P(B)
This formula expresses the idea that the probability of both events happening is the product of the probability of each event happening independently.
For example, if the probability of event A happening is 0.3 and the probability of event B happening is 0.4, then the probability of both events happening is 0.3 * 0.4 = 0.12
It’s important to note that independence is a property of the events, not the outcomes. For example, if you roll two dice, the outcome of the first roll does not affect the outcome of the second roll, so the events of rolling a 6 on the first die and rolling a 6 on the second die are independent.
Also it’s important to note that this formula only holds true when events A and B are independent, if they are not independent, then a different formula or approach should be used.
Independent Events Venn Diagram
A Venn diagram is a graphical representation of sets and their relationships. In the case of independent events, a Venn diagram can be used to visually depict the relationship between the two events and how their probabilities are calculated.
When two events A and B are independent, the probability of both events happening is the product of their individual probabilities. This is known as the multiplication rule for independent events. The formula is:
P(A and B) = P(A) * P(B)
In a Venn diagram, the sample space (the universal set) is represented by a large rectangle. The event A is represented by a circle inside the rectangle and the event B is represented by another circle inside the rectangle. The circles representing event A and event B do not overlap because they are independent events, meaning the outcome of one event does not affect the outcome of the other event.
The area of the intersection of the two circles represents the probability of both events happening, which is P(A and B) = P(A) * P(B). The area of the first circle represents the probability of event A happening, which is P(A) and the area of the second circle represents the probability of event B happening, which is P(B).
A Venn diagram can be a useful tool for visualizing the relationship between independent events and understanding how the probability of both events happening is calculated.
Independent events in probability example with solution
Here is an example of using the multiplication rule for independent events to calculate the probability of two events happening:
Example: A bag contains 6 red balls and 4 blue balls. A ball is drawn from the bag and then replaced, and then a second ball is drawn. What is the probability that both balls are red?
Solution: We can begin by using the multiplication rule for independent events. In this case, event A is drawing a red ball on the first draw, and event B is drawing a red ball on the second draw. Since we are replacing the ball after each draw, the events are independent.
The probability of event A happening is P(A) = 6/10, or 0.6. This is because there are 6 red balls out of a total of 10 balls in the bag.
The probability of event B happening is P(B) = 6/10, or 0.6. This is because there are still 6 red balls in the bag after the first draw, and the total number of balls is still 10.
Therefore, the probability of both events happening is P(A and B) = P(A) * P(B) = 0.6 * 0.6 = 0.36 or 36%.
So, the probability of drawing two red balls from the bag is 36%.
You can use the same reasoning to calculate the probability of drawing two blue balls, or one red and one blue ball.