s4w3 Discrete numbers, permutations, combinations

If I had a pair of trick dice and I calculate the experimental probability of rolling a sum of 11 as 500/1000 or 1/2. An unsuspecting mark would look at my dice and say the chance of getting an 11 is 2/36 or 1/18. There are still 36 different possible permutations of rolls with my two dice and two permutations that add to 11, why isn't the real theoretical probability not 2/36?

Vocab time! the set, or list, of possibilities for a probability is called the sample space. You will note that it is a set not a sequence. explain the difference between a set and a sequence?

Given fair dice, the larger the pool of trials, the closer the experimental probability should be to the theoretical probability

On a standard dice, what is the theoretical probability of rolling an even number?

If you want to find the probability of two different events both happening, you need to do what with the probabilities of each individual event happening

how many different ways are there to roll a 12 with two dice?

how many ways are there to roll a 6?

When you create the sample space for how to make a 6 with two dice, you will note that (2,4) is a different option than (4,2). If the order doesn't matter, this is called a combination. If the order does matter, it is called a permutation. You consider every permutation when determining the number of possible outcomes of a dice role. consider the possibilities if you roll two dice. You can see there are 36 permutations. How many combinations are there?

come up with two examples of when you need to figure out probabilities for dependent events (board games are a good source here)

come up with two examples of figuring out probabilities for two or more independent events.

How many different permutations are there for three dice rolled

Lets say you are randomly assigning roles for scenes from Romeo and Juliette in your class of 30. You are assigning 5 roles. how many entirely different casts could you possibly come up with? You start with the probability of choosing one of your 30 to start with. what is that probability?

you then consider what the probability is of picking any particular person for the second role. so - first is this second event dependent or independent?

what is the probability of choosing any particular person as the second role?

what is the probability of choosing any particular person as the third role?

What pattern are you seeing for dependent events?

For situations like this, you start seeing repeated multiplication where you are multiplying numbers that are counting down. You will note that this looks a little bit like sequences as series. Except, instead of adding, we are now multiplying, because when are finding the probability of separate events both happening, whether or not they are dependent, you multiply. So here is what that looks like.

This is called pi notation,( notice the capital pi, cause thats not confusing) . This doesn't really come up that often, and there is a convenient and simpler notation for the most common case called the factorial.

do you have any questions?

So you can see that if I wanted to know how many permutations I can have with 6 kids in 6 roles, the answer would be 6！ = 720

But what if I have 30 kids and 5 roles, which of the following would give me

?

sort which one is a combination or a permutation.

You know from binomial theorem, that the first term for

is

and the fourth term is

how is this read?

You know from binomial theorem, that the first term for

is

and the third term is

calculate

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