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JEE 2023 Mathematics PYQ — A coin is biased so that the head is 3 times as likely to occur a… | Mathem Solvex | Mathem Solvex

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JEE 2023 — Mathematics PYQ

JEE | Mathematics | 2023

A coin is biased so that the head is 3 times as likely to occur as tail. This coin is tossed until a head or three tails occur. If $X$ denotes the number of tosses of the coin, then the mean of $X$ is:

Choose the correct answer:

A.
$\frac{21}{16}$
(Correct Answer)
B.
$\frac{15}{16}$
C.
$\frac{81}{64}$

Correct Answer:

$\frac{21}{16}$

Explanation

Solution:

Given:

$P(H) = \frac{3}{4}$ and $P(T) = \frac{1}{4}$ .

Probability Distribution of $X$ :

For $X = 1$ : $P(X = 1) = P(H) = \frac{3}{4}$
For $X = 2$ : $P(X = 2) = P(TH) = \frac{1}{4} \times \frac{3}{4} = \frac{3}{16}$
For $X = 3$ : $P(X = 3) = P(TTH) + P(TTT) = (\frac{1}{4})^2 \times \frac{3}{4} + (\frac{1}{4})^3$

$P(X = 3) = \frac{3}{64} + \frac{1}{64} = \frac{4}{64} = \frac{1}{16}$

Mean Calculation:

\text{Mean} = \sum X \cdot P(X) = 1 \cdot \frac{3}{4} + 2 \cdot \frac{3}{16} + 3 \cdot \frac{1}{16}

\text{Mean} = \frac{3}{4} + \frac{6}{16} + \frac{3}{16} = \frac{12 + 6 + 3}{16} = \frac{21}{16}

Explanation

Solution:

Given:

$P(H) = \frac{3}{4}$ and $P(T) = \frac{1}{4}$ .

Probability Distribution of $X$ :

For $X = 1$ : $P(X = 1) = P(H) = \frac{3}{4}$
For $X = 2$ : $P(X = 2) = P(TH) = \frac{1}{4} \times \frac{3}{4} = \frac{3}{16}$
For $X = 3$ : $P(X = 3) = P(TTH) + P(TTT) = (\frac{1}{4})^2 \times \frac{3}{4} + (\frac{1}{4})^3$

$P(X = 3) = \frac{3}{64} + \frac{1}{64} = \frac{4}{64} = \frac{1}{16}$