What makes two events independent

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Example 2: A box contains 4 red marbles, 3 green marbles and 2 blue marbles. Subjects Near Me. Download our free learning tools apps and test prep books. In this case, the probabilities for the second pick are affected by the result of the first pick.

The events are considered to be dependent or not independent. Sa mpling with replacement: Suppose you pick three cards with replacement. The first card you pick out of the 52 cards is the Q of spades. You put this card back, reshuffle the cards and pick a second card from the card deck. It is the ten of clubs. You put this card back, reshuffle the cards and pick a third card from the card deck. This time, the card is the Q of spades again. You have picked the Q of spades twice.

You pick each card from the card deck. Sampling without replacement: Suppose you pick three cards without replacement. The first card you pick out of the 52 cards is the K of hearts. You put this card aside and pick the second card from the 51 cards remaining in the deck. It is the three of diamonds. You put this card aside and pick the third card from the remaining 50 cards in the deck. The third card is the J of spades.

Because you have picked the cards without replacement, you cannot pick the same card twice. Example 1 You have a fair, well-shuffled deck of 52 cards. Suppose you know that the picked cards are Q of spades, K of hearts and Q of spades. Can you decide if the sampling was with or without replacement? Show Answer Sampling with replacement. Show Answer No, we cannot tell if the sampling was with or without replacement.

Show Answer Without replacement With replacement. Try It You have a fair, well-shuffled deck of 52 cards. B and C are mutually exclusive. B and C have no members in common because you cannot have all tails and all heads at the same time. Try It Draw two cards from a standard card deck with replacement. F and G are not mutually exclusive. Getting all tails occurs when tails shows up on both coins TT. J and H are mutually exclusive. Try It A box has two balls, one white and one red.

Show Answer Yes. Show Answer No. Therefore, C and D are mutually exclusive events. No, event C and event E are not mutually exclusive events. Show Answer This is a conditional probability. Try It In a bag, there are six red marbles and four green marbles. S has ten outcomes. Try It A student goes to the library. Find P B D. 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. Now to find the probability, we observe that choosing from only the males in the school essentially alters the sample space from all students in the school to all male students in the school. The total number of possible outcomes is no longer , but has changed to In the above visual illustration, it is clear we are calculating a row percent.

The two-way table illustrates the idea via counts, while the Venn diagram converts the counts to probabilities, which are presented as regions rather than cells. We will want, however, to write our formal expression for conditional probabilities in terms of other, ordinary, probabilities and therefore the definition of conditional probability will grow out of the Venn diagram.

Since we know or it is given that the patient experienced insomnia , we are looking for P H I. Now that we have introduced conditional probability, try the interactive demonstration below which uses a Venn diagram to illustrate the basic probabilities we have been discussing. As we saw in the Exploratory Data Analysis section, whenever a situation involves more than one variable, it is generally of interest to determine whether or not the variables are related.

In probability, we talk about independent events , and earlier we said that two events A and B are independent if event A occurring does not affect the probability that event B will occur. Consider again the two-way table for all students in a particular high school, classified according to gender and whether or not they have one or both ears pierced. To answer this, we may compare the overall probability of having pierced ears to the conditional probability of having pierced ears, given that a student is male.

The probability of a student having pierced ears changes in this case, gets lower when we know that the student is male, and therefore the events E and M are dependent. Remember, if E and M were independent, knowing or not knowing that the student is male would not have made a difference … but it did.

The previous example illustrates that one method for determining whether two events are independent is to compare P B A and P B.