Let us consider some different real life examples and compute their respective probabilities. Consider this popular game Spin the bottle:
Q) What is the probability that the bottle lands at 180 degree angle on rotation?
Ans. The answer is unbelievably 0. The reason is that we can never calculate with complete exactness, the probability of a bottle landing at 180 or 180.00000 or 180.0000.
This shows us an interesting facet of continuous spaces, that the probability to compute anything in a continuous space is 0.
Let's see if we can answer the question after some modification.
Q) What is the probability that the bottle lands at any degree between 0 and 180?
Ans. It has become easier now. We have discretized the problem and compute the total sample space as any degree between 0 and 360. As 180 is half of 360, we can say that the required probability is 1/2.
Let's do it for some more ranges.
Q) What is the probability that the bottle lands at any degree between 20 and 150?
Ans. Now, the range has decreased. There are (150 -20) possible degrees at which a bottle can land out of a total 360 degrees. This makes the required probability:
Q) What is the probability that the bottle lands at any degree between 160 and 161?
Ans. This range has reduced tremendously. Applying the basic rules of probability, there is only 1(161 - 160 ) possible degrees out of a total of 360 degrees. Then, the computed probability is:
which is infinitesimally small!!
As we see the futility to calculate exact probabilities in a continuous space, we have to employ a new weapon from our arsenal. We call it density, which gives us an approximate estimate of the probability in a continuous space. We will now use powerful visualization tools like graphs to understand this concept.
Firstly, let us plot a graph with the possible degrees as x axis and the probability f(x) as y axis:
There are 2 important things to remember from the above plot:
- Every value in the range from 0 to 360 is assumed to have equal value, which can be computed as 1/360.
- The total area of the graph integrates to 1.
Another important thing to note here is that all the distributions we are considering are uniform, i.e. they are changing linearly.
Let us consider a case when the density is not uniform.
Q) What is the density of probability distribution graph of the birth given that the possibility of being born after noon is twice the probability of being born before noon?
Ans. The probability distribution graph can be drawn as:
Let us consider the density for birth to take place before noon as a and after noon as b.
We know the volume of this entire volume is 1. We can divide this into 3 volumes each
From the properties 1, we can see that each point has value 1/12 in the square between 1 and 12. Also, the volume of the particular rectangle with b as a height is 1/3. Hence , b is
Also, from figure and the information given, a = 2*b = 1/18.
- Probability of any point in a continuous space is 0.
- Density is an approximate value to the probability of a point in a continuous space.
3. In a density distribution graph, there are 2 important points to remember:
a) Every value in the range from 0 to 360 is assumed to have equal value, which can be computed as 1/360.
b) The total area of the graph integrates to 1.
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