Carl Horowitz

**University of Michigan**

Runs his own tutoring company

Carl taught upper-level math in several schools and currently runs his own tutoring company. He bets that no one can beat his love for intensive outdoor activities!

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When calculating the probability of multiple events, we must determine if the events are dependent or independent of one another. When calculating the **probability of dependent events** we must take into account the effect of one event on the other. An example of calculating the probability of dependent events is the probability of drawing two specific cards in a row with the second card being drawn from a smaller deck.

Often times we're finding the probability of events sometimes they are independent events where one event doesn't affect another one and often times they will also be dependent where one event does affect the other one. So we're going to look at this problem and sort of compare and contrast the difference between independent and dependent. Okay so what we're doing is we are grabbing marbles out of a bag we have 6 green and 4 blue and we're trying to find the probability you draw a green and then a blue.

Okay so the first thing we're going to do is what's called with replacement so we're taking out a marble and then we're putting it back in. Okay so the first draw we want to grab a green, we have 6 green marbles out of a 10 total. So the probability of drawing a green is going to be 6 out of 10 okay, we put that marble back because we're dealing with replacement and then we want to grab a blue. So when we grab our blue we have 4 blues out of 10 because we put it back. So now we just have a 4 tenth chance of that blue, to find the probability of both occurring we just multiply and end up with 24 over 100, that could be simplified but I'm not terribly concerned with the actual numeric value, just want the concept of what's going on. Okay so that is with replacement and those are independent events, it didn't matter that I drew this green first the probability of blue is still the same thing no matter what.

Okay without replacement so now I am taking a marble and I'm not putting it back in. Okay so we still need to grab that green first, there's still 10 marbles we're grabbing 1 there's 6 greens so the probability is still six tenth that we grab that green. We don't put that back so now instead of dealing with 10 marbles we're now dealing with 9. We want to grab a blue therefore a blue's probability is now just four ninths that we grab that blue multiply these probabilities together we end up with 24 out of 90. So our probability has increased because we didn't put that marble back. Okay, this is a dependent situation, this without replacement because where probability changes for the second marble depending on what happened with the first okay.

There are some formulas for dependent probability I tend to find them confusing, I tend to just sort of think about them logically and sort of look at the pool you're choosing from, first the outcome that you want. Okay, you can do a lot of these with t tree diagrams sort of diagrams okay what's going to happen here, here probability of each branch set off to another in general that will almost always give you the right answer it is just going to create a little bit more work than just thinking about it logically.

Okay so dependent events when one event varies the outcome of another just think about how they impact each other and how the probabilities change given that first event.