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Probability

Probability in math is the numerical representation of the chance of happening or not happening of an event. If I ask you, "What is the probability of getting a heads on flipping a coin?", you may answer "It is 1/2". Do you know how you got 1/2? Maybe not. If not, then let me explain:

There are two sides of a coin: A heads and a tails. When you flip a coin, you can get either heads or tails. So there are two possible 'outcomes' to the 'event' of throwing a coin (notice the terms outcomes and event). Out of these two possible outcomes, getting heads is just one. So the probability of getting heads becomes 1/2.

From the above discussion, we can conclude that probability of a particular outcome from an event is the number of outcomes that you want over the total number of outcomes that you can have. In a formula, this would appear as:
P("outcome") = "Number of outcomes you want"/"Total number of outcomes"
This is the formula for Theoretical Probability. Theoretical probability is called 'theoretical' because when we calculate it, we find out the probability before even performing the task; Do you have to throw the coin to know that the probability of getting heads is 1/2? No. You know that before you throw the coin. In other words, you calculated the probability without performing the task or experiment. This is Theoretical Probability.

Now let us actually perform an experiment. Let us take a fair coin (a fair coin means a coin that is not biased towards showing either heads or tails) having two sides, heads and tails, and throw it a thousand times (yes, you read that correct, a thousand times) and record the number of heads and tails we get.

Instead of going into the details of performing the experiment, let us see the results here:
• Number of tails = 495
• Number of heads = 505
We know that the probability of getting a heads is 1/2, but the above results show that out of one thousand throws, the coin showed heads 505 times, which is not exactly half of the number of trials. How can that be possible?

This is possible because Theoretical Probability is, well, theoretical. It is not always in accordance with the actual results of an experiment. If we were to determine the probability from the experiment above, then we would say that the probability of getting heads is 505 out of 1000, that is 505/1000 = 0.505 (which, again, is very near to 0.5). This probability is based on the actual results of the experiment and holds only for the experiment that you performed. It may not hold good for any other similar experiment. This kind of a probability is called Experimental Probability.

In dealing with probability problems, generally we do not perform the experiments stated in the problems to compute the probabilities, and hence, we generally only compute the Theoretical Probability in solving them.