An ordinary dice is rolled for a
Question:

An ordinary dice is rolled for a certain number of times. If the probability of getting an odd number 2 times is equal to the probability of getting an even number 3 times, then the probability of getting an odd number for odd number of times is :

  1. (1) $\frac{3}{16}$

  2. (2) $\frac{1}{2}$

  3. (3) $\frac{5}{16}$

  4. (4) $\frac{1}{32}$


Correct Option: , 2

Solution:

$\mathrm{P}$ (odd no. twice ) $=\mathrm{P}$ ( even no. thrice )

$\Rightarrow{ }^{n} C_{2}\left(\frac{1}{2}\right)^{n}={ }^{n} C_{3}\left(\frac{1}{2}\right)^{n} \Rightarrow n=5$

Success is getting an odd number then $P$ (odd successes) $=P(1)+P(3)+P(5)$

$={ }^{5} C_{1}\left(\frac{1}{2}\right)^{5}+{ }^{5} C_{3}\left(\frac{1}{2}\right)^{5}+{ }^{5} C_{5}\left(\frac{1}{2}\right)^{5}$

$=\frac{16}{2^{5}}=\frac{1}{2}$

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