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:
Correct Option: , 4
${ }^{\mathrm{n}} \mathrm{C}_{2}\left(\frac{1}{2}\right)^{\mathrm{n}}={ }^{\mathrm{n}} \mathrm{C}_{3}\left(\frac{1}{2}\right)^{\mathrm{n}} \Rightarrow{ }^{\mathrm{n}} \mathrm{C}_{2}={ }^{\mathrm{n}} \mathrm{C}_{3}$
$\Rightarrow \quad n=5$
Probability of getting an odd number for odd number of times is
${ }^{5} \mathrm{C}_{1}\left(\frac{1}{2}\right)^{5}+{ }^{5} \mathrm{C}_{3}\left(\frac{1}{2}\right)^{5}+{ }^{5} \mathrm{C}_{5}\left(\frac{1}{2}\right)^{5}=\frac{1}{2^{5}}(5+10+1)$
$=\frac{1}{2}$
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