# A die is thrown 5 times.Find the probability that an odd number will come up exactly three times.

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• Die is thrown 5 times In a single throw of die
• $P$(getting odd)$=\large\frac{3}{6}=\large\frac{1}{2}$
• $p=\large\frac{1}{2},$ $q=1-\large\frac{1}{2}$$=\large\frac{1}{2}$
• $$X=0\; X=1\; X=2\; X=3\; X=4\; X=5$$
• X={$$0\;1\;2\;3\;4\;5$$}
• $P(X=r)=^n C_r p^r q^{n-r}$
Die is thrown $5$ times. $\therefore \:n=5$
$P(X=$odd no comes exactly 3 times) $=P(X=3)$
We know that $P(X=r)=^n C_r p^r q^{n-r}$
$\Rightarrow\:P(X=3)=^5C_3\;(\frac{1}{2}\;)^3\;(\frac{1}{2}\;)^2$
$=10\times\large(\frac{1}{2^5})$
$=\large\frac{5}{16}$
edited Apr 7, 2014