# Probability Definition

Discussion in 'Probability and Statistics' started by nycmathguy, Jul 6, 2021.

1. ### nycmathguy

Joined:
Jun 27, 2021
Messages:
5,386
422
In general, the probability of an event happening = (number of ways it can happen)/(total number of outcomes).

What exactly does that mean?

nycmathguy, Jul 6, 2021
2. ### MathLover1

Joined:
Jun 27, 2021
Messages:
2,989
2,884
Probability associates numbers with the chances of a certain outcome happening, so that the higher that number, the greater the chance of that result occurring.

In other words, the probability of an event happening can be found by finding the ratio of the number of ways an event can happen and the number of possible outcomes.

Suppose an event “E” can happen in “r” ways out of a total of “n” possible equally likely ways.

Then the probability of occurrence of the event (called its success) is denoted by

P(E)= r/n

The probability of non-occurrence of the event (called its failure) is denoted by
_
P(E)= (n-r)/n=n/n-r/n=1-r/n

Notice the bar above the E, indicating the event does not occur.
_
Thus, P(E) +P(E)=1

MathLover1, Jul 6, 2021
nycmathguy likes this.
3. ### nycmathguy

Joined:
Jun 27, 2021
Messages:
5,386
422
This reply is a textbook reply. I am looking for a "Math For Dummies" definition. I found that the mathisfun site has a somewhat decent explanation but I wanted to see what you have to say about it. Be back with some very easy questions.

nycmathguy, Jul 6, 2021
4. ### nycmathguy

Joined:
Jun 27, 2021
Messages:
5,386
422
See below.

In general, the probability of an event happening = (number of ways it can happen)/(total number of outcomes).

"number of ways it can happen."

This means what we want, right?
What we want to take place, right?

"total number of outcomes."

This is the total number of items, people, events, books, etc, right?

nycmathguy, Jul 6, 2021
MathLover1 likes this.
5. ### MathLover1

Joined:
Jun 27, 2021
Messages:
2,989
2,884
right

MathLover1, Jul 6, 2021
nycmathguy likes this.
6. ### nycmathguy

Joined:
Jun 27, 2021
Messages:
5,386
422
Ok. I really want to learn probability well. Of course, we are also working on precalculus.

nycmathguy, Jul 6, 2021
7. ### MathLover1

Joined:
Jun 27, 2021
Messages:
2,989
2,884
concentrate on precalculus now, you will have a time to learn probability

MathLover1, Jul 6, 2021
nycmathguy likes this.
8. ### nycmathguy

Joined:
Jun 27, 2021
Messages:
5,386
422
Ok. I will put a hold on probability for now.

nycmathguy, Jul 6, 2021
MathLover1 likes this.
9. ### MathLover1

Joined:
Jun 27, 2021
Messages:
2,989
2,884
but , it's good to know probability specially if you are gamer
for example, when you know that
probability to hit jackpot is 1 in 13,983,816
or
1/13983816 which is approximately 0.00000007151 or 0.000007151%, you won't play

MathLover1, Jul 6, 2021
nycmathguy likes this.
10. ### nycmathguy

Joined:
Jun 27, 2021
Messages:
5,386
422
I am not a gamer. I like the algebra of probability. Nothing more; nothing less.

nycmathguy, Jul 7, 2021
11. ### Jonathan

Joined:
Oct 3, 2021
Messages:
9
3
There is before we know what the outcome was, and after we know what the outcome was.
There is also a set of all the possible outcomes that could have happened.

If we roll a fair six-sided die, there are six possible outcomes that could have happened.

If we roll a red die and a green die, there are 36 possible outcomes. Six where the red die was a 1 and the green die was any of it's possible outcomes, six more where the red die was 2 and the green die was any of it's six possible outcomes, and so on. 6x6 = 36 outcomes. Since these dice are declared to be a fair, each of the 36 outcomes will approach an equal number of appearances the more times we roll them.

But if we look instead at the total number of pips displayed, there are several outcomes that achieve the various totals. There is only one outcome that totals 2 {1,1}, but there are two that total three pips [{1,2}, {2,1}] and there are three outcomes that total four [ {1,3}, {2,2}, {3,1} ] and so on.

So we can arrange all the possible outcomes into sets that have a given total, and show them in a spin-the-wheel arrangement:

Jonathan, Oct 11, 2021
nycmathguy likes this.
12. ### nycmathguy

Joined:
Jun 27, 2021
Messages:
5,386
422
I thank you for taking the time to provide a detail reply.

nycmathguy, Oct 11, 2021
Jonathan likes this.
13. ### HallsofIvy

Joined:
Nov 6, 2021
Messages:
161
78
All of these answers should include that caution that this is assuming that every way an event can happen is "equally likely". For example you cannot say "If I walk across this street two things can happen- either I get hit by a car or I don't get hit by a car. Therefore the probability I will get hit by a car is 1/2!"

HallsofIvy, Nov 8, 2021
nycmathguy likes this.
14. ### nycmathguy

Joined:
Jun 27, 2021
Messages:
5,386
422
Cool. Good to know.

nycmathguy, Nov 8, 2021
15. ### HallsofIvy

Joined:
Nov 6, 2021
Messages:
161
78
Yes, it is good to know that, every time you walk across a street, the probability you will get hit by a car is NOT 1/2!

HallsofIvy, Nov 10, 2021

Joined:
Jun 27, 2021
Messages:
5,386