A die is rolled a certain number of times. If the probability of getting an odd number 2 times is same as the probability of getting an even number 3 times. The probability of getting an odd number an odd number of times is

Solution:

Let the die be rolled n times. It is given that

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

Let the required probability be p. Then,

$\begin{array}{l} p =^{5} C_{1} {(\frac{1}{2})}^{1} {(\frac{1}{2})}^{4} +^{5} C_{3} {(\frac{1}{2})}^{3} {(\frac{1}{2})}^{2} +^{5} C_{5} {(\frac{1}{2})}^{5} {(\frac{1}{2})}^{0} \\ \Rightarrow p = (^{5} C_{1} +^{5} C_{3} +^{5} C_{5}) \times \frac{1}{32} = \frac{2^{4}}{2^{5}} = \frac{1}{2} \end{array}$

Solution:

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