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 whether the events are dependent or independent. The probability of independent events is slightly less complicated than calculating the probability of dependent events. When calculating the **probability of independent events**, unlike the probability of dependent events, we do not need to take into account the effect of one event occurring on the other.

Finding the probability of independent events. So independent events are events that have no bearing on each other, okay if something happens over here, something happens over there as long as they're not affecting each other they're called independent events, and basically what we do is in order to find the probability of 2 events occurring independently we take the probability of one and multiply it by the probability of the other. So the probability of 2 things happening independently is just the probability of one times the probability of the other.

Okay one classic example for independent probability is basically pulling balls, marbles anything out of a bag okay. And so what we're looking at in this case is we have 2 bags containing marbles, bag a has 7 green and 6 black, bag b has 8 yellow and 4 red. We pick 1 marble from each bag probability that we choose a green and a red. Okay so choosing a marble out of bag a and then choosing a marble out of bag b are completely independent. It doesn't matter what happens out of the first bag, it could result from the second bag are independent from that.

Okay, so what we can do then is just find the probability of picking a green times the probability of a red. I should back up a second and just point out that no colors are duplicated so therefore it's not like we have to worry about having the green from a or b, there's really no overlap potential. Okay, bag a has green and black, bag b has yellow and red there's no overlap going on. Okay, so probability of choosing a green, the only greens we have are in bag a so we just need to find the probability of taking the green out of bag a. There are 7 green and 6 black so that tells me there's 7 green and 13 total. So the probability is just going to be 7 out of 13, the probability of choosing a red. The only reds occur in that second bag so then we have a 4 red out of 12 total chance of picking a red. And then we simply multiply these together 4 out of 12 can be simplified to one third and then we multiply across end up with 7 out of 39.

Okay, so long as you have independent events, events that have no bearing on each other all you have to do is multiply the probabilities together in order to find the probability of them both occurring.