Problem
A person randomly selects one of six envelopes. Each envelope contains a check that the person gets to keep. Determine the person's expectation if three envelopes contain a $\$ 539$ check and three envelopes contain a $\$ 1000$ check
Solution
Step 1 ：Define the total number of envelopes as 6.
Step 2 ：Define the number of envelopes containing a $539 check as 3.
Step 3 ：Define the number of envelopes containing a $1000 check as 3.
Step 4 ：Calculate the probability of selecting an envelope with a $539 check as \( \frac{3}{6} = 0.5 \).
Step 5 ：Calculate the probability of selecting an envelope with a $1000 check as \( \frac{3}{6} = 0.5 \).
Step 6 ：Calculate the expectation as \( 0.5 \times 539 + 0.5 \times 1000 = 769.5 \).
Step 7 ：The person's expectation is $769.5. So, the final answer is \( \boxed{769.5} \).
