A share of a company costs Rs 
500
500 today. Suppose today's share price increases by 50 % with probability 
0.6
0.6 and decreases by 50 % with probability 
0.4
0.4.Independent of today, suppose that tomorrow's share price increases by 20% with probability 
0.5
0.5, and decreases by 30% with probability 
0.5
0.5. If you decide to buy 
3
3 shares today, find the expected profit (in Rs) at the end of 2 days

To find the expected profit, we'll calculate the expected value of the stock price after 2 days and compare it to the initial investment.

### Step 1: Initial investment
The cost of 3 shares today is:
\[
\text{Initial Investment} = 3 \times 500 = 1500 \, \text{Rs}
\]

### Step 2: Share price at the end of Day 1
Let the share price increase by 50% with probability 0.6 or decrease by 50% with probability 0.4. After Day 1, the possible outcomes for the share price are:

- Increase by 50%: New price = \(500 \times (1 + 0.50) = 750 \, \text{Rs}\)
- Decrease by 50%: New price = \(500 \times (1 - 0.50) = 250 \, \text{Rs}\)

The expected share price at the end of Day 1 is:
\[
\text{Expected Price on Day 1} = 0.6 \times 750 + 0.4 \times 250 = 450 + 100 = 550 \, \text{Rs}
\]

### Step 3: Share price at the end of Day 2
Independent of Day 1, the share price on Day 2 can increase by 20% or decrease by 30%, both with probability 0.5. So, we need to calculate the expected share price for both possible outcomes of Day 1.

#### Case 1: Price after Day 1 is 750 Rs
- Increase by 20%: New price = \(750 \times (1 + 0.20) = 900 \, \text{Rs}\)
- Decrease by 30%: New price = \(750 \times (1 - 0.30) = 525 \, \text{Rs}\)

Expected share price if Day 1 ends with 750 Rs:
\[
\text{Expected Price} = 0.5 \times 900 + 0.5 \times 525 = 450 + 262.5 = 712.5 \, \text{Rs}
\]

#### Case 2: Price after Day 1 is 250 Rs
- Increase by 20%: New price = \(250 \times (1 + 0.20) = 300 \, \text{Rs}\)
- Decrease by 30%: New price = \(250 \times (1 - 0.30) = 175 \, \text{Rs}\)

Expected share price if Day 1 ends with 250 Rs:
\[
\text{Expected Price} = 0.5 \times 300 + 0.5 \times 175 = 150 + 87.5 = 237.5 \, \text{Rs}
\]

### Step 4: Overall expected price at the end of Day 2
Now, we combine the outcomes of Day 1 and Day 2 to find the overall expected share price at the end of Day 2:
\[
\text{Overall Expected Price} = 0.6 \times 712

A share of a company costs Rs 500 today. Suppose today's share price increases by 50% with probability 0.6 and decreases by 50% with probability 0.4. Independent of today, suppose that tomorrow's share price increases by 20% with probability 0.5 and decreases by 30% with probability 0.5. If you decide to buy 3 shares today, find the expected profit (in Rs) at the end of 2 days.

Find the expected profit when a company's stock price fluctuates over two days, considering both probability and percentage changes. Learn how to apply probability concepts and expected value calculations in real-world scenarios. Suitable for Grades 10-12.

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