We begin with a verbal definition of independent events later we will use probability notation to define this more precisely. She randomly extracts one of the coins and, after looking at it, replaces it before picking a second coin. Let Q1 be the event that the first coin is a quarter and Q2 be the event that the second coin is a quarter. Since the first coin that was selected is replaced , whether or not Q1 occurred i.
In either case whether Q1 occurred or not , when she is selecting the second coin, she has in her pocket:. She randomly extracts one of the coins, and without placing it back into her pocket, she picks a second coin. As before, let Q1 be the event that the first coin is a quarter, and Q2 be the event that the second coin is a quarter.
Since the first coin that was selected is not replaced, whether Q1 occurred i. If Q1 occurred i. However, if Q1 has not occurred i.
In these last two examples, we could actually have done some calculation in order to check whether or not the two events are independent or not. Sometimes we can just use common sense to guide us as to whether two events are independent.
Here is an example. Let B1 be the event that one of the people has blue eyes and B2 be the event that the other person has blue eyes. In this case, since they were chosen at random, whether one of them has blue eyes has no effect on the likelihood that the other one has blue eyes, and therefore B1 and B2 are independent. Let B1 be the event that one child has blue eyes, and B2 be the event that the other chosen child has blue eyes.
In this case, B1 and B2 are not independent, since we know that eye color is hereditary. Thus, whether or not one child is blue-eyed will increase or decrease the chances that the other child has blue eyes, respectively. The idea of disjoint events is about whether or not it is possible for the events to occur at the same time see the examples on the page for Basic Probability Rules. The idea of independent events is about whether or not the events affect each other in the sense that the occurrence of one event affects the probability of the occurrence of the other see the examples above.
The following activity deals with the distinction between these concepts. The purpose of this activity is to help you strengthen your understanding about the concepts of disjoint events and independent events, and the distinction between them. Now that we understand the idea of independent events, we can finally get to rules for finding P A and B in the special case in which the events A and B are independent.
Later we will present a more general version for use when the events are not necessarily independent. Since they were chosen simultaneously and at random, the blood type of one has no effect on the blood type of the other. Therefore, O1 and O2 are independent, and we may apply Rule The purpose of this comment is to point out the magnitude of P A or B and of P A and B relative to either one of the individual probabilities.
Since probabilities are never negative, the probability of one event or another is always at least as large as either of the individual probabilities. Since probabilities are never more than 1, the probability of one event and another generally involves multiplying numbers that are less than 1, therefore can never be more than either of the individual probabilities. Modify it to a more general event — that a randomly chosen person has blood type A or B — and the probability increases.
Modify it to a more specific or restrictive event — that not just one randomly chosen person has blood type A, but that out of two simultaneously randomly chosen people, person 1 will have type A and person 2 will have type B — and the probability decreases. Before we move on to our next rule, here are two comments that will help you use these rules in broader types of problems and more effectively.
In fact, they are equally likely. The idea here is that the probabilities of certain events may be affected by whether or not other events have occurred. All the students in a certain high school were surveyed, then classified according to gender and whether they had either of their ears pierced:. Note that this is a two-way table of counts that was first introduced when we talked about the relationship between two categorical variables.
It is not surprising that we are using it again in this example, since we indeed have two categorical variables here:. Since a student is chosen at random from the group of students, out of which are pierced,. Since a student is chosen at random from the group of students out of which 36 are male and have their ear s pierced,.
At this point, new notation is required, to express the probability of a certain event given that another event holds. Example: your boss to be fair randomly assigns everyone an extra 2 hours work on weekend evenings between 4 and midnight. What are the chances you get Saturday between 4 and 6?
Example: the chance of a flight being delayed is 0. Result: 0. Example: you are in a room with 30 people, and find that Zach and Anna celebrate their birthday on the same day. Do you say: "Wow, how strange! Why is the chance so high? Because you are comparing everyone to everyone else not just one to many. And with 30 people that is comparisons Read Shared Birthdays to find out more. Example: Snap! Did you ever say something at exactly the same time as someone else?
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