The 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 :   Option: 1 $\frac{3}{16}$ Option: 2 $\frac{1}{32}$ Option: 3 $\frac{5}{16}$ Option: 4 $\frac{1}{2}$

$\\\mathrm{P}(\text { odd no. twice })=\mathrm{P}(\text { even no. thrice }) \\ \\\Rightarrow{ }^{n} \mathrm{C}_{2}\left(\frac{1}{2}\right)^{\mathrm{n}}={ }^{n} \mathrm{C}_{3}\left(\frac{1}{2}\right)^{\mathrm{n}} \Rightarrow \mathrm{n}=5$

\begin{aligned} &\text { success is getting an odd number then } \mathrm{P}(\text { odd successes })=\mathrm{P}(1)+\mathrm{P}(3)+\mathrm{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} \end{aligned}

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