A coin is tossed K times. If the probability of getting 3 heads is equal to the probability of getting 7 heads, then the probability of getting 8 tails is:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(\displaystyle\frac{45}{1024}\)
The number of heads in K tosses of a fair coin is binomial with p = q = 1/2:
\( \displaystyle P(r \text{ heads}) = {}^{K}C_{r}\left(\frac{1}{2}\right)^{K} \)
The given condition becomes
\({}^{K}C_{3} = {}^{K}C_{7}\)
Since \({}^{n}C_{r} = {}^{n}C_{s}\) with \(r \ne s\) means r + s = n:
\(K = 3 + 7 = 10\)
In 10 tosses, 8 tails means 2 heads:
\( \displaystyle P = {}^{10}C_{2}\left(\frac{1}{2}\right)^{10} = \frac{45}{1024} \)