Question

In a game, a man wins \( ₹1000 \) if he gets an even number greater than or equal to 4 on a fair dice and loses \( ₹200 \) for getting any other number on the dice. If he decides to throw the dice until he wins or maximum of three times, then his expected gain/loss in \( ₹ \) is -----------

• A. \( \frac{2200}{9} \) loss
• B. \( \frac{3800}{9} \) gain
• C. \( \frac{2200}{9} \) gain
• D. \( \frac{3800}{9} \) loss

Hint:

In expected value questions, list every possible stopping case with its probability and corresponding gain or lossThen multiply and add.

Answer 1

The Correct Option is B

Step 1: Find probability of winning in one throw.
Winning numbers are even numbers greater than or equal to 4.
So favourable outcomes are:
\[ 4,6 \]
Hence:
\[ P(\text{win}) = \frac{2}{6} = \frac{1}{3} \]

Step 2: Find probability of losing in one throw.
\[ P(\text{loss}) = 1-\frac{1}{3} = \frac{2}{3} \]

Step 3: List possible cases.
The man throws until he wins or maximum of three times.
Winning on first throw:
\[ P = \frac{1}{3} \]
Winning on second throw:
\[ P = \frac{2}{3}\cdot \frac{1}{3} = \frac{2}{9} \]
Winning on third throw:
\[ P = \frac{2}{3}\cdot \frac{2}{3}\cdot \frac{1}{3} = \frac{4}{27} \]
Losing all three throws:
\[ P = \left(\frac{2}{3}\right)^3 = \frac{8}{27} \]

Step 4: Find gains or losses in each case.
If he wins on first throw, gain is:
\[ 1000 \]
If he wins on second throw, he loses \(200\) once and wins \(1000\):
\[ 1000-200 = 800 \]
If he wins on third throw, he loses \(200\) twice and wins \(1000\):
\[ 1000-400 = 600 \]
If he loses all three throws:
\[ -600 \]

Step 5: Compute expected value.
\[ E = 1000\left(\frac{1}{3}\right) + 800\left(\frac{2}{9}\right) + 600\left(\frac{4}{27}\right) - 600\left(\frac{8}{27}\right) \]

Step 6: Simplify.
\[ E = \frac{1000}{3} + \frac{1600}{9} + \frac{2400}{27} - \frac{4800}{27} \]
\[ E = \frac{9000}{27} + \frac{4800}{27} + \frac{2400}{27} - \frac{4800}{27} \]
\[ E = \frac{11400}{27} \]
\[ E = \frac{3800}{9} \]

Step 7: Final conclusion.
Since expected value is positive, it is a gain.
\[ \boxed{\frac{3800}{9}\text{ gain}} \]