Probability

Sometimes you want to form sets of objects from a larger population of objects and sometimes the order of those objects matters! For instance, if you have a pile of 10 beads and you intend to make bracelets that use three different beads, but you consider each permutation (blue, red, purple and purple, blue, red) different, then you want to find how many permutations you can create.

Sometimes you want to form sets of objects from a larger population of objects.  For instance, if you are trying to create a team of 5 players, from a set of 10 players who try out, how many different teams can you make?  Or, you have 25 t-shirts, but can only pack 5 for a trip.  How many different groups of 5 shirts can you make?  In these cases, you want to use the combination formula.

Combinations get trickier when you don’t have to choose one item from each group.  If you are ordering lunch and can choose among three sandwiches, three types of chips, and three drinks, and you must choose one of each, you multiply your options together and get 27 possible combinations (for more, see the lesson on Combinations (Making sets, choosing one of each option)).

Finding simple combinations when merging two sets of events looks tricky, but it’s not.  As long as you are required to use one of each item, all you have to do is multiply all of the different options in one set times all of the different options of another set.

Why does this work?

When we think about probability, we usually think about cases in which we know how common an event is, and we want to know the odds it will happen in a particular instance.

For example, we know that a coin toss will turn up heads 50% of the time. So, when we flip a coin, we have a 50% chance of getting heads.

Or, we know that there are 4 aces in a deck of cards, so if we pull a random card from a deck, we have a $\dfrac{4}{52}$ chance of pulling an ace.

Probabilities show the chances (in percent or fraction form) of something happening (if that event is happening at random).  Probabilities are usually given in simplest form.  So, the odds of rolling a 5 on a die is $\dfrac{1}{6}$.

We can use these probabliies to predict how often something SHOULD happen if you do it a certain number of times.  We can also use probabilities to work backwards and figure out actual numbers in the scenario you have a probability for.

When you want to find the probability of multiple events happening, your probability that ALL events will happen is always going to be smaller than the probability that only one of the events will happen. So, whenever you have an AND probability, you want to combine the probability of each individual event into one total probability (which will always be smaller than the smallest individual probability).

Often, when we are trying to find the probability of something happening, we want to find the probability of several events happening. To find the probability of multiple events, we have to combine the probability that each event will happen.  Exactly how we combine the probabilities depends on exactly what we're trying to figure out!

Before we get into the rules of probability, let's think logically about how different types of questions might affect the way we calculate probabilities.

Most calculations dealing with probabilities assume that the probabilities are independent. In other words, they assume that the results of the first event do not affect the other events. Flipping two different coins is a great example of two independent events. We might be superstitious and think that if we get heads on the first coin, we're more likely to get tails on the second flip. But, in fact, each coin flip is totally independent.

Probability shows the frequency or odds in which an random events should, theoretically, occur.

So, if you flip a coin, you have a $\dfrac{1}{2}$ probability of getting heads. See the lesson on Probability (single events) for more details. 

In probability, "complementary events" are events that, if one occurs, the other definitely does not.  Complementary events are also sometimes called "mutually exclusive" events.