The Triple $L$ investment club is considering purchasing a certain stock. After considerable research, the club members determine that there is a $50\mathrm{\%}$ chance of making $\mathrm{\$}10,000$, a $10\mathrm{\%}$ chance of breaking even, and a $40\mathrm{\%}$ chance of losing $\mathrm{\$}6,200$. Find the expectation of this purchase.

The expected value is $\mathrm{\$}◻$.

Step 1 ：The Triple L investment club is considering purchasing a certain stock. After considerable research, the club members determine that there is a 50% chance of making $10,000,a10$6,200. We are asked to find the expectation of this purchase.

Step 2 ：The expectation of a random variable is the sum of the products of each outcome and its probability. In this case, the outcomes are making $10,000,breakingeven(making$0), and losing $6,200. The probabilities are 50%, 10%, and 40% respectively.

Step 3 ：We can calculate the expectation by multiplying each outcome by its probability and then summing these products. The outcomes are $10,000,$0, and -$6,200. The probabilities are 0.5, 0.1, and 0.4 respectively.

Step 4 ：The expectation is calculated as follows: $0.5\ast 10000+0.1\ast 0+0.4\ast (-6200)=2520$

Step 5 ：Final Answer: The expected value is $\boxed{{\displaystyle 2520}}$