Question

Vasant and Jothi play a game with a coin. Vasant stakes INR 1 and throws the coin four times. If he throws four heads, he gets his stake and INR3 from Jothi. If he throws only three heads and they are consecutive, he gets his stake and INR 2 from Jothi. If he throws only two heads and they are consecutive, he gets his stake and INR 1 from Jothi. In all other cases Jothi takes the stake money. Find the expectation of Vasant's gain.

• A. \( -\frac{5}{8} \)
• B. 0
• C. 1
• D. \( \frac{5}{8} \)

Hint:

For expected gain problems, list each possible gain with its probability and then use \(E(X)=\sum x_i p_i\).

Answer 1

The Correct Option is B

Step 1: Find total outcomes.
A coin is tossed four times, so total possible outcomes are:
\[ 2^4 = 16. \]

Step 2: Case of four heads.
Four heads occur only in one way:
\[ HHHH. \]
So, probability is:
\[ \frac{1}{16}. \]
Gain in this case is \(INR3\).

Step 3: Case of exactly three consecutive heads.
Exactly three heads and they are consecutive can occur as:
\[ HHHT,\quad THHH. \]
So, number of favourable outcomes is 2.
Probability is:
\[ \frac{2}{16}. \]
Gain in this case is \(INR2\).

Step 4: Case of exactly two consecutive heads.
Exactly two heads and they are consecutive can occur as:
\[ HHTT,\quad THHT,\quad TTHH. \]
So, number of favourable outcomes is 3.
Probability is:
\[ \frac{3}{16}. \]
Gain in this case is \(INR1\).

Step 5: Find all other cases.
Total favourable winning outcomes are:
\[ 1+2+3=6. \]
So, remaining outcomes are:
\[ 16-6=10. \]
In all these cases, Vasant loses his stake, so gain is:
\[ -1. \]

Step 6: Calculate expected gain.
\[ E(X)=3\left(\frac{1}{16}\right)+2\left(\frac{2}{16}\right)+1\left(\frac{3}{16}\right)+(-1)\left(\frac{10}{16}\right). \]
\[ E(X)=\frac{3}{16}+\frac{4}{16}+\frac{3}{16}-\frac{10}{16}. \]
\[ E(X)=\frac{10-10}{16}=0. \]

Step 7: Final conclusion.
Therefore, the expectation of Vasant's gain is 0.
Final Answer:
\[ \boxed{0} \]