Example: 5  
A fair coin is tossed times. Find the probability of obtaining

Solution: 5(a)  
Consider any arbitrary sequence of tosses that contains exactly Heads. For example, consider
Any such sequence has a probability of occurrence equal to times Thus, what we need to do is count the number of sequences with exactly Heads. The required probability will then be (No. of sequences).
Counting such sequences is a simple & problem, and those familiar with that subject will immediately hit upon the answer: there are such sequences.
Thus,
In general, we see that

Solution: 5(b)  
For this question, we consider all the possible cases and add their respective probabilities like this:

Solution: 5(c)  
Similarly, we have
This questions was an example of Binomial Distributions, something we’ll study in more detail later in this chapter.

Example: 6  
Consider the sequence of numbers . A person chooses three numbers at random from this sequence. Find the probability that the three numbers form an

Solution: 6  
Note that s can be formed with varying common differences . and are examples of s with . is an of . The maximum possible is , in the .
Let us count all such s in a table. Verify the column on the right.
Thus, the total number of s possible from this set is . Also, from this set of numbers, a selection of numbers can be made in ways.
Therefore, the probability that three numbers picked at random from this set form an is .
