An event that is dependent on preceding events is called a dependent occurrence. The outcomes of prior events have an impact on these occurrences. Two or more interdependent occurrences are defined as dependent events. If one occurrence is changed by chance, the chances are that another will be changed as well.
Dependent Events
Probability theory is an essential idea that most students who study mathematics in upper grades are familiar with. For instance, the weather forecast for some areas predicts that it will rain tomorrow with a 20% chance.
The probability is the chance of something happening. The term “event” refers to single or several results. The consequence that is possible is described as the event. Total occurrences are defined as all possible outcomes related to the experiment asked about in the inquiry. Favourable events are sometimes known as events of interest. In the branch of probability, an event is defined as the collection of all possible outcomes for a given experiment.
For example
- Obtaining ahead in a coin toss could be considered an event.
- Getting a 6 on a die roll is considered an event.
- Taking an ace from a deck of cards is likewise a special occasion.
- Getting a sum of 9 on a pair of dice is a rare occurrence.
Important Result:
When two events, A and B, are interdependent, the probability of occurrence of A and B is:
P(A and B) = P(A) · P(B|A)
For instance:
- i) Assume that three cards are to be drawn from a deck. When the first card is pulled, the chances of obtaining an ace are highest, whereas the chances of getting an ace are lower when the second card is drawn. This likelihood would be determined by the results of the previous two cards in the third card draw. We can say that after drawing one card, the deck will have fewer cards accessible, hence the probability will change.
- ii) Let’s pretend that we desire a queen. The chances of getting a queen with the first card drawn are 4 out of 52. If we obtain a queen in the first draw, we have a 3 out of 51 chance of getting a queen in the second draw. These are known as dependent events because the likelihood of the second event is determined by the outcome of the first.
The Probability of Dependent Events
If A and B are dependent events, then the probability of A and B occurring is written as:
Given, that the probability of event A is P(A)
Event B has a probability of P. (B after A)
P(B and A)=P(A)×P(B after A).
P(B after A) can also be written as P(B|A).
P(B|A) means that event A has already happened.
Now, what is the chance of happening in event B?
The “Conditional Probability” of B given A is also known as P(B|A).
Then P(B and A)=P(A)×P(B|A).
The Difference Between Independent and Dependent Events
Probability has two categories of events that are commonly referred to as dependent and independent.
In the following table, you can observe the difference.
Dependent Events | Independent Events |
---|---|
1. The likelihood of another event is affected by the occurrence of one event. | 1. The likelihood of another event is not affected by the occurrence of one. |
2. A power outage if you don’t pay your bill on time, or winning the lottery after purchasing 10 lottery tickets are examples (the more the tickets bought, the greater the chance of winning). | 2. Cycling and watching your favourite film are two examples of this. |
3. Formula can be written as: | 3. The formula can be written as: |
Finding Dependent Event Probabilities
The conditional probability formula is used to calculate the likelihood of dependent events:
If the probabilities of occurrences A and B are P(A) and P(B), the conditional probability of event B if event A has already occurred is P(B/A).
Calculate conditional probability using this formula.
P(BA)=P(A∩B)P(A)or P(B∩A)P(A)
P(A) must be bigger than zero.
If P(A) is smaller than 0, A is an improbable event. The intersection in P(A B) signifies a compound probability of an occurrence.
Checking Whether the Probability Belongs to Dependent or Independent Events in a Few Steps
Step 1: Is it feasible for the events to take place in any order? If so, proceed to step 2, otherwise, proceed to step 3.
Step 2: Does one occurrence have an impact on the result of the other? If so, proceed to step 4, otherwise, proceed to step 3
Step 3: The event is Self-Contained Simply plug in the independent event formula to get the answer.
Step 4: The event is conditional. Simply enter the dependent event formula to obtain the answer.
This is how you may determine whether an event is dependent or independent!
Simple dependent event examples include
- Robbery of a bank and subsequent imprisonment.
- Having your power turned off for not paying your power bill on time.
- Getting on the plane first and getting a decent seat.
- Getting a parking ticket for parking illegally. You’re more likely to obtain a ticket if you park illegally.
- Winning the lottery after purchasing ten lottery tickets. The more tickets you purchase, the better your chances of winning become.
- Being involved in a traffic accident while driving a car
For Example
A juggler uses seven red, five green, and four blue balls are used by a juggler. He drops a ball while performing his trick and fails to pick it up. Another ball falls as he continues. What are the chances that the first ball dropped was blue and the second ball was green?
Solution
The juggler does not replace the first ball, as we all know. After dropping the first ball, he now has 15 balls to play with.
P(blue ball) = 4/16 is the probability that the first ball will be blue.
P(green ball) = 5/15 is the probability that the second ball will be green.
The chances of the first ball being blue and the second ball being green are
P(bluer than green) = P(bluer than green)P (green) P(blue than green) = P(blue)P(green) =(416)(515)=1/12 P(blue than green)
Conclusion
In probability, dependent events are events whose occurrence affects the probability of the occurrence of the other event. Assume a bag contains three red and six green balls. Two balls are drawn one after the other from the bag. Let A represent the first draw of the red ball and B represent the second draw of the green ball. If the ball drawn in the first draw is not replaced in the bag, then A and B are dependent events because P(B) is reduced or increased depending on whether the first draw results in a red or green ball.