Problem:
There are 5 tickets in a box numbered 1, 1, 2, 2, 2 respectively. Two tickets are taken at random from 8. Find the expectation of the total of the numbers on the tickets.

Solution:

To find the expectation of the total of the numbers on the tickets, we need to calculate the expected value.

Step 1: Define the Random Variables
Let X and Y be the random variables representing the numbers on the first and second tickets drawn, respectively.

Step 2: Determine the Sample Space
Since there are 8 tickets in total, the sample space consists of all possible pairs of tickets that can be drawn. Therefore, the sample space is given by:
S = {(1, 1), (1, 2), (1, 2), (1, 2), (2, 1), (2, 2), (2, 2), (2, 2)}

Step 3: Assign Probabilities
Assign probabilities to each outcome in the sample space. Since the tickets are drawn at random, each outcome is equally likely. Therefore, the probability of each outcome is 1/8.

Step 4: Define the Function
Let Z = X + Y be the random variable representing the total of the numbers on the tickets.

Step 5: Calculate the Expected Value
The expected value of Z, denoted as E(Z), is given by the formula:
E(Z) = Σ (Z * P(Z))

To calculate E(Z), we need to calculate Z * P(Z) for each outcome in the sample space and sum them up.

Calculations:
Let's calculate Z * P(Z) for each outcome and sum them up.

(Z * P(Z)) = (2 * 1/8) + (3 * 2/8) + (3 * 2/8) + (3 * 2/8) + (3 * 1/8) + (4 * 2/8) + (4 * 2/8) + (4 * 2/8)

Simplifying the expression, we get:
(Z * P(Z)) = (2/8) + (6/8) + (6/8) + (6/8) + (3/8) + (8/8) + (8/8) + (8/8)

(Z * P(Z)) = 47/8

Therefore, the expected value of Z, E(Z), is given by:
E(Z) = Σ (Z * P(Z)) = Σ (47/8) = 47/8

Answer:
The expectation of the total of the numbers on the tickets is 47/8.