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14.Probability
hard
Let a die be rolled $n$ times. Let the probability of getting odd numbers seven times be equal to the probability of getting odd numbers nine times. If the probability of getting even numbers twice is $\frac{ k }{2^{15}}$, then $k$ is equal to:
A
$30$
B
$90$
C
$15$
D
$60$
(JEE MAIN-2023)
Solution
$P$ (odd number 7 times) $= P$ (odd number 9 times)
${ }^{ n } C _7\left(\frac{1}{2}\right)^7\left(\frac{1}{2}\right)^{ n -7}={ }^{ n } C _9\left(\frac{1}{2}\right)^9\left(\frac{1}{2}\right)^{ n -9}$
${ }^{ n } C _7={ }^{ n } C _9$
$\Rightarrow n =16$
Required
$P ={ }^{16} C _2 \times\left(\frac{1}{2}\right)^{16}$
$=\frac{16 \cdot 15}{2} \times \frac{1}{2^{16}}=\frac{15}{2^{13}}$
$\Rightarrow \frac{60}{2^{15}} \Rightarrow k =60$
Standard 11
Mathematics
