Multiple choice

A coin is so biased that the probability of falling head when tossed is $\displaystyle \frac{1}{4}$ . If the coin is tossed $5$ times the probability of obtaining $2$ heads and $3$ tails, with heads occurring in succession is

  1. $\displaystyle \frac{5\times 3^{3}}{4^{5}}$
  2. $\displaystyle \frac{3^{3}}{5^{4}}$
  3. $\displaystyle \frac{3^{3}}{4^{4}}$
  4. $\displaystyle \frac{3^{3}}{4^{5}}$
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C Correct answer
Explanation

Probability of head p=1/4, tail q=3/4. We need 2 heads and 3 tails with heads in succession. Possible patterns: HHTTT, THHTT, TTHHT, TTTHH. There are 4 such patterns. Each pattern has probability (1/4)^2 * (3/4)^3 = 27/1024. Total probability = 4 * 27/1024 = 27/256 = 3^3 / 4^4.

AI explanation

We want exactly 2 heads occurring in succession, which gives 4 valid positions for the block of heads: (HHTTT), (THHTT), (TTHHT), and (TTTHH). For each of these 4 arrangements, the probability is (1/4)^2 * (3/4)^3. Multiplying the 4 arrangements by this probability gives 4 * (1/16) * (27/64) = 27/256. Simplifying the fraction 27/256 yields 3^3 / 4^4.

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