Question:
A fair coin is tossed a fixed number of times. If the probability of getting 4 heads is equal to the probability of getting 7 heads, then the probability of getting 2 heads is
एक निष्पक्ष सिक्के को एक निश्चित संख्या में उछाला जाता है। यदि 4 शीर्ष (heads) आने की प्रायिकता 7 शीर्ष आने की प्रायिकता के बराबर है, तो 2 शीर्ष आने की प्रायिकता क्या है?
Options:
- \(\dfrac{3}{4096}\)
- \(\dfrac{1}{1024}\)
- \(\dfrac{3}{1024}\)
- \(\dfrac{55}{2048}\)
- \(\dfrac{3}{4096}\)
- \(\dfrac{1}{1024}\)
- \(\dfrac{3}{1024}\)
- \(\dfrac{55}{2048}\)
Correct Answer: d)
Explanation:
\(P(X=4) = P(X=7) \implies {}^nC_4 (1/2)^n = {}^nC_7 (1/2)^n \implies {}^nC_4 = {}^nC_7 \implies n = 4 + 7 = 11\).
\(P(X=2) = {}^{11}C_2 (1/2)^{11} = \dfrac{11 \times 10}{2} \times \dfrac{1}{2048} = \dfrac{55}{2048}\).
\({}^nC_4 = {}^nC_7 \implies n = 4 + 7 = 11\)।
\(P(X=2) = {}^{11}C_2 (1/2)^{11} = \dfrac{55}{2048}\)।