Why do we take the probability of an event ratio as favorable / possible outcome

In summary, the probability of an event is calculated by taking the number of favorable outcomes and dividing it by the total number of possible outcomes. This definition allows for the probability to be represented as a ratio and ensures that the probability of all possible outcomes adds up to 1. This approach is used to determine the likelihood of an event occurring in a given sample space.
  • #1
Juwane
87
0
Why do we take the "probability of an event" ratio as favorable / possible outcome

If A is an event, then probability that A will occur is given by

P(A)= no. of favorable outcomes / total no. of possible outcomes

Is this just a definition, or is there some special significance in taking the number of favorable outcomes as the numerator and total no. of possible outcomes as the denominator? Is it because since the numerator will always be less than the denominator, it will be easier to work with the ratio?
 
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  • #2


Definitions.

Given a sample space, the probability of the entire sample space is 1, the probability of any element in the sample space is [0,1].
 
  • #3


farful said:
Given a sample space, the probability of the entire sample space is 1, the probability of any element in the sample space is [0,1].

And how does this relate to the numerator being the number of favorable outcomes and the denominator being the total number of possible outcomes?
 
  • #4


Juwane said:
And how does this relate to the numerator being the number of favorable outcomes and the denominator being the total number of possible outcomes?

"Favorable" is a judgment within some context. A probability simply relates the number of designated outcomes to all possible outcomes. The designated outcomes could be favorable, unfavorable or neutral depending on your point of view.
 
  • #5


My question is that why the favorable outcomes is in the numerator and total possible outcomes is in the denominator? Why not vice versa? Is it just a definition or does it have a significance?
 
  • #6
Hi Juwane! :smile:

Because for two independent events, you want both P(A + B) = P(A) + P(B), and P(AB) = P(A)P(B), and favourable/possible is the only way of doing that. :wink:
 
  • #7


I still don't understand. I like to understand through an example:

A coin is tossed. The probability that we will get a head is 1 over 2.

Another example:

A die is cast. The probability that we will get a 3 is 1 over 6.

Now my question is why, in the first example, it is 1 over 2 and not 2 over 1; and why in the second example it is 1 over 6 and not 6 over 1?
 
  • #8
Juwane said:
… why in the second example it is 1 over 6 and not 6 over 1?

You could define it as 6 over 1, but then the probability of a 4 would be 6 over 1 also. and the probability of a 4 or a 3 would be 6 over 2, = 3 over 1.

And the probability of throwing the dice twice and getting a 4 first time and a 3 second time would be 36 over 1.
 
  • #9


tiny-tim said:
You could define it as 6 over 1, but then the probability of a 4 would be 6 over 1 also. and the probability of a 4 or a 3 would be 6 over 2, = 3 over 1.

And the probability of throwing the dice twice and getting a 4 first time and a 3 second time would be 36 over 1.

So it's just the definition, right? Because it makes sense to say that the probability of getting a head in a coin throw is half, so it's 1/2, right?
 
  • #10
Juwane said:
So it's just the definition, right? Because it makes sense to say that the probability of getting a head in a coin throw is half, so it's 1/2, right?

Right. :smile:

Mathematicians can define anything they like.

But a definition, even if perfectly valid, is only useful, if it successfully models the real world. :wink:
 
  • #11


Well, you could take instead favorable outcomes / unfavorable outcomes and call it "odds".

So odds of 10 : 1 mean two probabilities of 10/11 and 1/11.
 
  • #12


By Gee!, by Gosh! It was Laplace who DEFINED the probability of an event as the favorible outcomes divided by the total outcomes! It is the basic definition.
 
  • #13


Example: What is the probability that heads will fall on one throw of a coin? From theoretical probability, P(heads)=1/2. This means that 1/2 or 50% of your tosses of a coin should land on heads, theoretically. The numerator must be the favorable outcomes and the denominator must be the total possible outcomes. This ratio is 1 to 2 or 1:2, meaning, 1 out of every 2 tosses of the coin should result in heads. If you had possible outcomes in the numerator the ratio would be 2 : 1 , which would mean that you could toss 2 heads in 1 throw with one die. Impossible.
 
  • #14


bcramton said:
Example: What is the probability that heads will fall on one throw of a coin? From theoretical probability, P(heads)=1/2. This means that 1/2 or 50% of your tosses of a coin should land on heads, theoretically. The numerator must be the favorable outcomes and the denominator must be the total possible outcomes. This ratio is 1 to 2 or 1:2, meaning, 1 out of every 2 tosses of the coin should result in heads. If you had possible outcomes in the numerator the ratio would be 2 : 1 , which would mean that you could toss 2 heads in 1 throw with one die. Impossible.

2 possible outcomes doesn't mean they are both heads.
 
  • #15


Juwane said:
A die is cast. The probability that we will get a 3 is 1 over 6.

Now my question is why... is it 1 over 6 and not 6 over 1?

1] What would it accomplish to have it defined as 6/1? Are there any ways this would be better?

2] There are a number of very useful reasons for defining it as 1/6. For example, you can add up all the possbilities and arrive at 1 again: 1/6+1/6+1/6+1/6+1/6+1/6 = 1. i.e. all individual possibilities add up to the total possible outcomes.

How would you do this if it were represented as 6/1?
 

1. Why do we use probability as a ratio of favorable to possible outcomes?

The use of probability as a ratio of favorable to possible outcomes allows us to quantify the likelihood of an event occurring. By comparing the number of desired outcomes to the total number of possible outcomes, we can determine the likelihood of a specific outcome happening.

2. How does probability help us in decision making?

Probability helps us in decision making by providing us with a way to estimate the likelihood of a particular outcome. This information can be used to make informed decisions and assess potential risks and benefits.

3. What is the difference between theoretical and experimental probability?

Theoretical probability is based on mathematical calculations and assumes an ideal situation, while experimental probability is based on actual observations and may vary due to random factors. Theoretical probability is used to predict outcomes, while experimental probability is used to analyze past events.

4. Can probability be used to accurately predict the future?

No, probability cannot be used to accurately predict the future. It is a tool for estimating likelihood based on past events, but it does not guarantee that a specific outcome will occur in the future.

5. How can we use probability to improve our understanding of the world around us?

By studying probability, we can gain a better understanding of how likely certain events are to occur. This can help us make more informed decisions, assess risks, and better interpret data. Probability also plays a crucial role in many fields, such as economics, finance, and science, allowing us to make predictions and draw conclusions from data.

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