Lesson FURTHER - Probability

Source code of 'FURTHER - Probability'

This Lesson (FURTHER - Probability) was created by by longjonsilver(2297)

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About longjonsilver: I have a new job in September, teaching

Introduction
This is the second of my Lessons on Probability. It will expand upon the basics, covered in the first Lesson "BASICS - Probability".

EXAMPLE:

2 dice are rolled and their numbers added together. Find the following probabilities for the total on a single roll of the 2 dice:

1. P(getting 3)
2. P(getting 7)
3. P(getting 12)
4. P(getting 15)

The best way forward here is to create a matrix of the possible outcomes. Let the top number be one die and let the vertical number at the ledt be the other die. The numbers inside are the totals.

+  1 2 3  4  5  6
-------------------
1   2 3 4  5  6  7
2   3 4 5  6  7  8
3   4 5 6  7  8  9
4   5 6 7  8  9 10
5   6 7 8  9 10 11
6   7 8 9 10 11 12

so now we have something visual to look at. Note there are 36 different possible answers.

solutions:
1. P(getting 3) = 2 outcomes out of 36 --> 2/36 --> 1/18
2. P(getting 7) = 6 outcomes out of 36 --> 7/36 
3. P(getting 12) = 1 outcomes out of 36 --> 1/36 
4. P(getting 15) = 0 outcomes out of 36 --> 0/36 --> 0

EXAMPLE:
A coin is tossed and a die is rolled. Find the following probabilities:

1. P(getting a head and a 5)
2. P(getting a tail with either a 1 or a 2)

Again, draw a table:

H1 H2 H3 H4 H5 H6
  T1 T2 T3 T4 T5 T6

solution:
1. P(H5) = 1 outcome from 12 --> 1/12
2. P(getting a tail with either a 1 or a 2) = 2 outcomes from 12 --> 2/12 --> 1/6

When to Add and when to Multiply probabilities
From a mathematical point of view, when outcomes are mutually exclusive, we ADD. When they are independent, we MULTIPLY.

So, what does this actually mean? Well, Think about the 2 dice question about.

What is the probability of "rolling a 1 on the first die and then rolling a 3 on the second"? We know the answer from writing out the table above: P(1 and then a 3) = 1/36 since there is only one way of doing this.

Lets look at this question more closely...
P(1 and then a 3) = P(1) AND then P(3)

We know P(1) = P(3) = 1/6 each. I also know that AND is multiply, so:

P(1) AND then P(3) = 1/6 * 1/6 
P(1) AND then P(3) = 1/36.

--> This is the answer we got just by looking at the table.

However, what if the question was worded ever so slightly differently as "Find the probability of rolling a 1 and a 3". Now this differs from my first question in just one word...no "then". This is very important to how i understand the question since now we have:

P(1 and 3) = [P(1) AND then P(3)] OR [P(3) AND then P(1)]

Can you see how asking the question slightly differently now allows for both possibilities of rolling the "1 then the 3" OR the "3 then the 1". Both are valid outcomes.

P(1 and 3) = [P(1) AND then P(3)] OR [P(3) AND then P(1)]
P(1 and 3) = [1/6 * 1/6] + [1/6 * 1/6]
P(1 and 3) = [1/36] + [1/36]
P(1 and 3) = 2/36
P(1 and 3) = 1/18

I think these 2 questions show the subtly in probability questions. The actual maths is not difficult, but understanding what is being asked is the difficulty. You have to be careful!

Here is another example, from the website
*[problems 26742/14538]