Tree Diagram

Have any of you heard of a tree diagram? Or what it is used for? Well we learned about them today and well they were confusing at first, but actually really fun after I understood how they worked. Here is an example of a Tree Diagram. They are used to help calculate probabilities and helps you understand how it works. So I hope this picture doesn't scare you right off the back because it is actually easier than it looks. 

Lets do one that will go along with this graph so you can have a visual aid. Suppose there is vending bouncy ball machine. In this machine there are 3 Red Balls, 2 Green Balls, and 1 White Ball. There is this mother named Sally who has two kids, twins to be exact and she wants to get them each a ball. What is the probability of getting a red, and then a green ball in that order?

So in the FIRST STAGE

On the line going up to red we would put 3/6
On the line that goes to the white one we would put 1/6
Finally on the line that goes to the green we would write 2/6

Now if you were to add up all the numerators you should get 6
1+2+3=6

Now the SECOND STAGE would be written like this, since there is one ball being taken out each time we have to take that into account.

Red to Red = 2/5
Red to White = 1/5
Red to Green = 2/5

Add the numerators up and you get 5

White to Red = 3/5
White to White = 0/5
White to Green = 2/5

Add the numerators up and you get 5

Green to Red = 3/5
Green to White = 1/5
Green to Green = 1/5

Add the numerators up and you get 5

Now for the THIRD STAGE you have

RR 6/30 + GG 2/30 = 8/30 = 4/15

The probability of Sally getting a red and then a green ball is 4/15

Whats the Difference Between Probability and Odds?

So far this semester this was one of the hardest things for me to understand was the difference between probability and odds. After looking at my test that was over this I clearly can see I was not entirely grasping this concept. So to help you all out I am going to try to explain it so you can have a better understanding than I did. To just make sure we are on the same page, I didn't understand it because I was taught wrong, it is because it was a difficult concept for me. So I want to share what I know to help you out. Here are a couple of sites that might be of some use to use while learning the differences between probability and odds.
                          - Changing Probability to Odds 
                          - Differences Between Odds and Probability
                          - Finding the Odds
                          - Probability of Events

Lets start by simple definition of each

Probability: Is the amount of successes divided by the total.

Odds in Favor: Is the amount of successes divided by the amount of failures.

Odds Against: Is the amount of failures divided by the amount of successes.

An example of probability would be if I was to say I have a 60% chance of getting a banana out of a basket that had only 6 bananas and 4 apples.  I would have a 6/10 = 3/5 chance

Odds in Favor: Would be 3:2 to get that I need to make sure that when the two numbers add up that they equal the total. So I subtract the successes from the total to give me the failures.

5-3=2 Since 3 is the successes and 2 is the failures and Odds in Favor is the amount of successes over amount of failures we get this...

3:2

Odds Against: Would be 2:3, and since we know that Odds Against is amount of failures over amount of successes it write it like this...

2:3

Rock-Paper-Scissors

Have you ever played Rock, Paper, Scissors? Have you ever felt that your opponent knew what you where going to choose, therefore your opponent always choosing the thing that will make you lose? Is this a fair game? What are the chances of you winning? Losing? Or even tying? Well before we go into how this all works here is a video that should help you understand a little more about how competitive this game really is...

Ok so now that we have that out of the way, lets learn how we can actually find out if this game is a fair or rigged game. Today in class we actually found out if it was a fair game or not, and here is how we did it.
First we got into groups of two and played  45 games of Rock-Paper-Scissors and kept track who won and lost with rock paper and scissors, and we also kept track of the ones we tied. Afterwards we added up (individually) how many times I won, my partner won, and how many times we tied. In my example I won 19/45, Tied 10/45, Lost 16/45 times. We then were asked to explain if it was fair or not. We both decided it was because There was not a huge margin between the wins. So theoretically it is a fair game. Here is the reason why....

If you were to make a 3x3 square and mark the first column and row with Paper, then the second column and row with Scissors, then finally the last column and row with Rock. Now Player A will be the column side and Player B will the row side. Fill out your 3x3 square with all the possible outcomes and mark where there would be a tie with T, if player A would win mark it with an A, likewise for player B but with B.
What is the probability of player A winning? = 3/9 = 1/3
What is the probability of player B winning? = 3/9 = 1/3
What is the probability of a tie? = 3/9 = 1/3

Here is a link to that will also provide another way of seeing if the game is fair or not.
         Is Rock-Paper-Scissors Fair?

You now can see that Rock-Paper-Scissors is a fair game.