ONE MORE STEP TOWARD BETTER TOMORROW : chapter-10 - Worked Out Examples

Thursday, 7 August 2014

chapter-10 - Worked Out Examples – 3

Example: 5

A fair coin is tossed $10$ times. Find the probability of obtaining

	(a) exactly $6$ Heads
	(b) at most $6$ Heads
	(c) at least $6$ Heads

Solution: 5-(a)

Consider any arbitrary sequence of $10$ tosses that contains exactly $6$ Heads. For example, consider

${\rm{\{ H\, T\, H\, H\, H \,T \,T \,H \,H\, T\} }}$

Any such sequence has a probability of occurrence equal to $\dfrac{1}{2} \times \dfrac{1}{2} \times \dfrac{1}{2}\ldots 10$ times $= \dfrac{1}{{{2^{10}}}}$ Thus, what we need to do is count the number of sequences with exactly $6$ Heads. The required probability will then be $\dfrac{1}{{{2^{10}}}} \times$ (No. of sequences).

Counting such sequences is a simple $P$ & $C$ problem, and those familiar with that subject will immediately hit upon the answer: there are ${}^{10}{C_6}$ such sequences.

Thus,

	$P ({\rm{exactly\,}} 6 \,{\rm{Heads}}){ = ^{10}}{C_6} \times \dfrac{1}{{{2^{10}}}}$
	$= \dfrac{{210}}{{1024}}$
	$= \dfrac{{105}}{{512}}$

In general, we see that

$P ({\rm{exactly}} \,n\,{\rm{ Heads}}) = \dfrac{{^{10}{C_n}}}{{{2^{10}}}}$

Solution: 5-(b)

For this question, we consider all the possible cases and add their respective probabilities like this:

	${\rm{P(at\, most\, 6 \,Heads) = P(0\, Head)+ P(1\, Head) + P(2\, Heads}})$ $+ P(3 \,{\rm{Heads) + P(4 \,Heads) + P(5\, Heads) + P(6\, Heads)}}$
	$= \dfrac{{^{10}{C_0}}}{{{2^{10}}}} + \dfrac{{^{10}{C_1}}}{{{2^{10}}}} + \dfrac{{^{10}{C_2}}}{{{2^{10}}}} + \dfrac{{^{10}{C_3}}}{{{2^{10}}}} + \dfrac{{^{10}{C_4}}}{{{2^{10}}}} + \dfrac{{^{10}{C_5}}}{{{2^{10}}}} + \dfrac{{^{10}{C_6}}}{{{2^{10}}}}$
	$= \dfrac{{1 + 10 + 45 + 120 + 210 + 252 + 210}}{{1024}}$
	$= \dfrac{{53}}{{64}}$

Solution: 5-(c)

Similarly, we have

	${\rm{P(at\, least\, 6\, Heads) = P(6 \,Heads) + P(7 \,Heads) + P(8 \,Heads) + P(9 \,Heads) + P(10\, Heads)}}$
	$= \dfrac{{^{10}{C_6}}}{{{2^{10}}}} + \dfrac{{^{10}{C_7}}}{{{2^{10}}}} + \dfrac{{^{10}{C_8}}}{{{2^{10}}}} + \dfrac{{^{10}{C_9}}}{{{2^{10}}}} + \dfrac{{^{10}{C_{10}}}}{{{2^{10}}}}$
	$= \dfrac{{210 + 120 + 45 + 10 + 1}}{{{2^{10}}}}$
	$= \dfrac{{193}}{{512}}$

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 $1,2,3,\ldots ,13$ . A person chooses three numbers at random from this sequence. Find the probability that the three numbers form an $A.P.$

Solution: 6

Note that $A. P.$ s can be formed with varying common differences $(CD)$ . $(1, 2, 3)$ and $(3, 4, 5)$ are examples of $A. P.$ s with $CD$ $1$ . $(2, 5, 8 )$ is an $A. P$ of $CD$ $3$ . The maximum $CD$ possible is $6$ , in the $A. P.\,\, (1, 7, 13)$ .

Let us count all such $A. P.$ s in a table. Verify the column on the right.

	$CD=1 \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,{\rm{No.\, of \, A.P.s}}$
	$1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,11$
	$2\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,9$
	$3\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,7$
	$4\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,5$
	$5\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,3$
	$6\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,1$
	${\rm{Total}}=36$

Thus, the total number of $AP$ s possible from this set is $36$ . Also, from this set of $13$ numbers, a selection of $3$ numbers can be made in ${}^{13}{C_3} = 286$ ways.

Therefore, the probability that three numbers picked at random from this set form an $A.P.$ is $\dfrac{{36}}{{286}} = \dfrac{{18}}{{143}}$ .

Thursday, 7 August 2014

chapter-10 - Worked Out Examples – 3

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