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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Sometimes we have a system of equations that has either infinite or zero solutions. We call these **no solution systems of equations**. When we solve a system of equations and arrive at a false statement, it tells us that the equations do not intersect at a common point. One scenario is that 2 or more of the planes are parallel or that two of the planes intersect and the other intersects at a different point.

When we're dealing with the probability of multiple events what we have to look at is if our events mutually exclusive meaning there's no overlap or if they're inclusive meaning there is overlap. So what we're going to do is take a look at a couple of problems and see the difference between when we know when something overlaps and when something doesn't. So for both situations what we're going to do is we're going to be rolling a die and we're going to be dealing with the probability of rolling 2 different things one or the other.

For our first case what we're looking at is rolling a 5 or a even number okay, so what you'll sometimes see is instead of writing it out as a sentence, just say the probability of a 5 or even and for this one I just want to sort of write out what our potential outcomes could be. Okay so obviously we roll a 5 that's going to work, so we could roll a 5 or an even number so our potential for even numbers are just 2, 4 and 6. So we have 4 numbers that we're concerned with that are going to fit this mould. Out of 6 potential sides to a dice, so basically we have 4 numbers that work over 6 and we end up with a 2 over 3 probability we get either a 5 or an even number.

Okay so that's dealing with no overlap, there's no way to have a 5 that is even. The other situation is to have overlap and so for this one what we're looking at is a 5 or an odd number okay. So to roll a 5 obviously we can just roll a 5 and you're on odd what we do is we could roll a 1 hope if I'd known the difference between even and odd 1, 3 or 5. So before what we did was to just add the number here and the number here. But there's overlap now the number 5 appears both in the 5 and the odd categories so we can't count it twice because we would just double count it. So really the only numbers we're concerned with are 1, 3 and 5 leaving us with just one half. So the general formulas that we're looking at is basically if we have mutually exclusive events, if we have events that don't overlap what we do is we add the probability of 1 and the other.

So the probability of a or b as long as they don't overlap is just a probability of a plus the probability of b. If they are inclusive or overlap the formula is very similar we still have the probability of a plus the probability of b but what we have on the end is subtracting the probability that they both occur okay. Let's go take a look at this last problem and see how this formula works. So what we get is the probability that we roll a 5 probability of a 5 is just going to be one sixth okay plus the probability of an odd which is going to be 3 out of 6 minus the probability of 5 and odd okay. Any 5 by default is odd so the chance of getting 5 and odd is just the probability of rolling a 5 which is just one sixth.

What will end up happening is the one sixth plus 3 sixth minus one sixth, one sixth cancels up giving us one half. Okay the formula in this case was a little bit harder because it's a pretty straight forward problem but as we get into more difficult problems it's really convenient to use this formula because each individual thing maybe a little bit easier, there maybe more of an overlap which can cause a little bit more chaos if we're trying to write it all out. Okay so finding the probability of multiple events just consider if they overlap or not, if your results are going to sort of influence each other and then just use the appropriate formula.