Question

An ordinary dice is rolled 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:

• A. $\frac{1}{32}$
• B. $\frac{5}{16}$
• C. $\frac{1}{2}$
• D. $\frac{3}{16}$

Answer 1

The correct option is C $\frac{1}{2}$

Let the no. of throws be n.
Probability of getting an odd no. twice = ${}_2^{n}C\ast {\left(\frac{1}{2}\right)}^n$
Probability of getting an even no. thrice = ${}_3^{n}C\ast {\left(\frac{1}{2}\right)}^n$
Since the above are equal, we get: ${}_2^{n}C{=}_3^{n}C=>n=5$
Probability of getting an odd number once or thrice or 5 times = ${(}_1^{5}C{+}_3^{5}C{+}_5^{5}C)\ast {\left(\frac{1}{2}\right)}^5=\frac{1}{2}$
Hence, (c) is correct.