Introduction:
To find the expectation of the total value of the numbers on the three coins, we need to consider all the possible outcomes and their respective probabilities.

Step 1: Understanding the possible outcomes:
When three coins are tossed, there are 2^3 = 8 possible outcomes. Let's represent the heads as 'H' and tails as 'T'. The possible outcomes are:
1. HHH (total value = 1+1+1 = 3)
2. HHT (total value = 1+1+2 = 4)
3. HTH (total value = 1+2+1 = 4)
4. THH (total value = 2+1+1 = 4)
5. HTT (total value = 1+2+2 = 5)
6. THT (total value = 2+1+2 = 5)
7. TTH (total value = 2+2+1 = 5)
8. TTT (total value = 2+2+2 = 6)

Step 2: Calculating probabilities:
Since each coin has 2 possible outcomes (H or T) and the coins are tossed independently, the probability of each outcome is 1/2 * 1/2 * 1/2 = 1/8.

Step 3: Calculating the expectation:
The expectation is calculated by multiplying each outcome with its probability and summing them up. In this case, it is the sum of the total values multiplied by their probabilities.

Expected Value = (3 * 1/8) + (4 * 3/8) + (5 * 3/8) + (6 * 1/8)
= 3/8 + 12/8 + 15/8 + 6/8
= 36/8
= 4.5

Conclusion:
The expectation of the total value of the numbers on the three coins is 4.5. This means that if we perform this experiment multiple times, the average value of the outcomes will be 4.5.