SOLUTION: My Algebra I book has this question. I know how to solve it, but I want to understand WHY the formula works for finding probability. Here's the problem: "When you play a game

Question 335676: My Algebra I book has this question. I know how to solve it, but I want to understand WHY the formula works for finding probability. Here's the problem:
"When you play a game with number cubes, you can find probabilities by squaring a binomial. Let A represent rolling 1 or 2 and B represent rolling 2,3,4, or 6. The probability of A is 1/3 and the probability of B is 2/3. Find the solution: (1/3A + 2/3B)squared."
Again, I want to understand WHY the formula (1/3A + 2/3B)squared works for identifying probability. Thank you!!

Answer by stanbon(75887) (Show Source): You can put this solution on YOUR website!
When you play a game with number cubes, you can find probabilities by squaring a binomial. Let A represent rolling 1 or 2 and B represent rolling 2,3,4, or 6. The probability of A is 1/3 and the probability of B is 2/3. Find the solution: (1/3A + 2/3B)squared."
Again, I want to understand WHY the formula (1/3A + 2/3B)squared works for identifying probability.
------------------------------
[(1/3)A + (2/3)B]^2
----
= [(1/3)A]^2 + 2[(1/3)A(2/3)B] + [(2/3)B]^2
--------------
= (1/9)A^2 + 2(1/3)(2/3)AB + (4/9)B^2
--------------
= (1/9)AA + 2(1/3)(2/3)AB + (4.9)BB
--------------
This says you can get the event AA one way and the probability is (1/3)(1/3)
-------------------------
You can get the event AB or BA in two ways and the probability is (1/3)(2/3)
---------------------------
You can get the event BB in 1 way and the probability is (2/3)(2/3)
---------------------------------------
That perfectly describes all the possible results when you roll
a die twice and it shows you the probability of the different outcomes.
---------------------------------------
You ask why?
I don't think you can ask that of a mathematical model.
The only question you can ask is "How good is the model?"
In this case the model is perfect so it is used.
--------------------
Note ((1/3)A+(2/3)B)^n would show you all the results of rolling
your die n times and give you the probability of each result.
--------------------
This model was discovered by Blaise Pascal. The model he developed
is called "Pascal's Triangle"
====================================
Cheers,
Stan H.

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