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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These are actually going to be dealing with independent events, because whatever name is drawn in the first place gets marked, and then put back in, shaken up, drawn again. So the first name doesn't necessarily affect the second name. So therefore we know we have independent events.

The probability of independent events is just a probability of one event times the probability of another. So we're going to do some problems with this set up. Basically it's 30 students, and we're drawing two names with replacements. So we're drawing a name, and putting it back. We're replacing that first name.

So let's look at a couple of different problems. The first problem I want to consider is the probability that you are picked twice. So basically what we're going to do is a probability you're picked the first times, and then the probability you're picked the second time, and we have to multiply those together.

You are just one person, and there are 30 of you in your class, so basically the probability that you are picked is 1 out of 30.So that's the probability the first time you are picked. The second time, you're name has been put back in, so it's going to be exactly the same. So this is just 1 out of 30 times 1 out of 30 which is just going to be 1 out of 900.

The probability that you aren't picked. So this means that somebody else, it doesn't matter who, but somebody else is picked both times. So you're 1 which means there's 29 other people in your class. So the probability of somebody else being picked that first time is 29 out of 30. Their name is put back. There is once again 30 students that you're picking from. Again, you only have a 1 out of 30 chance of winning, so you have a 29 out of 30 chance of not getting picked. So you end up with 29² over 900. 29² I'm not really concerned with the number just the process that we're going through.

The last one we're looking at is the probability you are picked once. So we have to consider two things. We have to consider you being picked first, or you being picked second.

The probability that you are picked first. You have a 1 out of 30 chance of being picked first times that second time, someone else has to get picked. So there is a 29 out 30 chance that they're picked.

Similarly these orders could be reversed. Someone else could be picked first, and then you could be picked which is going to be the same ratio as this, but what we're going to have to go is multiply this by 2.

Another way of looking at that is you are being picked once out of 2. So we have to multiply this 2 choose 1 which is just 2, but either way this is going to need a multiplied by 2, because you could either be picked first of second, because order doesn't necessarily matter for the outcome, but it's a different event if it happens first or second.

So dealing with independent events; pulling a name out of a hat, putting back in, pulling another name out. Those are independent events. One outcome doesn't affect the other. So when you find the probability, all you have to do is multiply the probability of one times the probability of the other.