ONE MORE STEP TOWARD BETTER TOMORROW : chapter- 19 Worked Out Examples

Thursday, 7 August 2014

chapter- 19 Worked Out Examples

Example: 17

Find the $PD$ of the number of doublets in three throws of a pair of dice.

Solution: 17

In three throws, we can obtain either $0$ , $1$ , $2$ , or $3$ doublets. Denote this $RV$ by $X$ .

Now, doublets can be obtained in $6$ ways, namely $(1, 1)$ , $(2, 2)$ , $(3, 3)$ , $(4, 4)$ , $(5, 5)$ and $(6, 6)$ , out of a total possible ways of $6 \times 6 = 36$ . Thus, the probability of obtaining a doublet is

	${p_1} = P({\rm{doublet}}) = \dfrac{1}{6}$
	and ${p_2} = P({\rm{Not\, a \,Doublet}}) = \dfrac{5}{6}$

Now, let us obtain the probabilities for the different values of $X$ .

	$X = 0 \Rightarrow\,\, P(X = 0) = P$ {No doublets in all three throws}
	$= \dfrac{5}{6} \times \dfrac{5}{6} \times \dfrac{5}{6}$
	$= \dfrac{{125}}{{216}}$
	$X = 1 \Rightarrow \,\,P(X = 1) = P$ {A doublet exactly once}
	$= \left\{ {\dfrac{1}{6} \times \dfrac{5}{6} \times \dfrac{5}{6}} \right\} \times 3$ {Why is there the factor of $3$ ?}
	$= \dfrac{{75}}{{216}}$

$X = 2 \Rightarrow P(X = 2) = P$ {Exactly two doublets}

	$= \left\{ {\dfrac{1}{6} \times \dfrac{1}{6} \times \dfrac{5}{6}} \right\} \times 3$ (why a factor of $3$ again?)
	$= \dfrac{{15}}{{216}}$

$X = 3 \Rightarrow\,\, P(X = 3) = P$ {Doublets in all three throws}

	$= \dfrac{1}{6} \times \dfrac{1}{6} \times \dfrac{1}{6}$
	$= \dfrac{1}{{216}}$

Thus, the $PD$ is

Note the (obvious) fact that $\sum\limits_{i = 0}^3 {P(X = i) = 1}$

Example: 18

Find the $PD$ of the number of Heads in four tosses of a coin.

Solution: 18

Let $X$ denote the number of Heads in the four tosses

Verify the following probabilities:

	$P(X = 0) = \dfrac{1}{{{2^4}}} = \dfrac{1}{{16}}$
	$P(X = 1) = {\,^4}{C_1} \times \dfrac{1}{{{2^4}}} = \dfrac{1}{4}$
	$P(X = 2) = {\,^4}{C_2} \times \dfrac{1}{{{2^4}}} = \dfrac{3}{8}$
	$P(X = 3) = {\,^4}{C_3} \times \dfrac{1}{{{2^4}}} = \dfrac{1}{4}$
	$P(X = 4) = \,\dfrac{1}{{{2^4}}} = \dfrac{1}{{16}}$

Thus, the $PD$ is

Thursday, 7 August 2014

chapter- 19 Worked Out Examples

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