"The various possible ways are decided using permutation or combination. But what is the possibility that a arrangement can occur is called probability.. Randomness in this world requires probability to predict..!! All game plays are made with some defined principles and randomness..But how to measure at least well defined randomness so that to predict something? plainly with probability"See Also
In mathematics, you have deterministic or non-deterministic problems. All non-deterministic problems are called probabilistic problems. Simply, All math done with formula's are deterministic..!! Even for generating randomness, we use formula's. But note that these are non-deterministic formulas!!.. You can never say when they will fail!!.. a formula should never fail right ? :)
The beauty of probability is to check how much we can pass/fail rather than simply pass!!.. It is sometimes just a condition over permutations/combination. That is, reason we studied about permutation & set theory well before.
Pass/Fail means Probability.. Which means.. Pass = 1, Fail = 0. So the value of probability will lie between 0 and 1.
If probability of some event is 0, then it means it will never occur (utter fail or flop). If its 1, then it will definitely occur(success).
What is the event a probability measure? By now, you would have guessed. Random and known time interval occurring events. But we need to put some question for which we need to calculate probability (it can be either % failure or % success calculation).
The common example, flipping coin. The probability of getting tail 50%. The probability of getting head is 50%. Because the event is equally likely. You will either get a tail or head with equal probability.
But not all events are equally likely like this. Lets take an example question like this.
A couple has 2 kids and atleast one is a boy. What is the probability that both are boys?
Get the permutations of 2 kids: GG, BG, GB, BB.. 4 arrangements. Since we only care about probability of getting 2 boys, Ignore GG. Total 3 arrangements in which we find a boy out of which only 1 has both of them as boys.
So the probability is: 1/3
Some more math about Probability Point-by-Point.
- Probability values are always between 0 and 1. So, it also like 0% possible, 100% possible
- It operates over a sample space called Experiment(E). It's nothing but the set of all the permutations. The set of permutation changes based on the kind of probability question we ask!!
- We call some thing as event if it occurs once in the experiment. Event is nothing but subset in the Experiment set. We calculate the probability of an Event in the experiment!!
- All set theory concepts applies to our experiment set(sample space). You can combine two experiments (Union). You can take common events across Experiments(Intersection). You can even take difference between two experiments.
- Disjoint sets as we already studied means empty set when you take intersection between them. Same case applies here, disjoint events means no relation between the two events.
- If you have two events and you know that always 1 event occurs, then we can use conditional probability. Conditional probability is nothing but taking intersection between the two events. (take randomness of only common events)
Sampling is a statistical concepts. Probability too works with samples. There are many more mathematics part of statistics like differential equations and integrals. Distributions are all defined with differentials and integrals. Distributions are basically used to generalize the function. The same distributions can be used to generalize a probability function as a random variable. This is called Probability distributions.
Whatever probability you have, the probability distribution runs from 0 to 1 and also there will be a mean value between these two numbers.
Example: If there are large number of trails of tossing the coin, then heads will be the max with probability around 50%.
Where it is used ?
Probability and expected value function is a must to know to analyze the random algorithms. We studied all this just because we are going to study randomized algorithms :).
It is known that the randomized algorithms perform really well in average case for specific problems.
More about Probability
I would suggest referring a better math book rather than me writing about it. There are many more formulas and most used probability functions. Please refer a proper book.