Problem Analysis:

The problem involves 8 counters, each marked with specific numbers:

• Counters: {1, 1, 2, 4, 4, 5, 5, 5}

Fiona randomly selects three counters and calculates their total. We are tasked with finding the probability that the total of these three counters is odd.

Key Idea:

The sum of three numbers is odd if and only if:

• 1 odd number + 2 even numbers, OR
• 3 odd numbers.

We analyze these cases step-by-step.

Step 1: Classification of Odd and Even Numbers

From the given counters:

• Odd Numbers: {1, 1, 5, 5, 5} → 5 odd counters
• Even Numbers: {2, 4, 4} → 3 even counters

Step 2: Total Possible Combinations

The total number of ways to select 3 counters from 8 is given by the combination formula $\binom{n}{r}$, where $n = 8$ and $r = 3$:

$\binom{8}{3} = \frac{8 \cdot 7 \cdot 6}{3 \cdot 2 \cdot 1} = 56$

So, there are 56 total combinations.

Step 3: Odd Sum Cases

Case 1: 1 Odd + 2 Even

• To select 1 odd number from the 5 odd counters: $\binom{5}{1} = 5$
• To select 2 even numbers from the 3 even counters: $\binom{3}{2} = 3$

The total number of combinations for this case is: $\binom{5}{1} \cdot \binom{3}{2} = 5 \cdot 3 = 15$

Case 2: 3 Odd Numbers

• To select 3 odd numbers from the 5 odd counters: $\binom{5}{3} = \frac{5 \cdot 4 \cdot 3}{3 \cdot 2 \cdot 1} = 10$

The total number of combinations for this case is: $10$

Step 4: Total Favorable Outcomes

Adding the two cases together, the total number of favorable outcomes is: $15 + 10 = 25$

Step 5: Probability Calculation

The probability is the ratio of favorable outcomes to total outcomes: $P(\text{odd sum}) = \frac{25}{56}$

Final Answer:

The probability that Fiona's total is odd is: $\boxed{\frac{25}{56}}$

Would you like me to break down the steps further or clarify any part? Let me know!

5 Follow-Up Questions:

1. What is the probability that Fiona’s total is even instead?
2. How many ways are there to select exactly 2 odd counters and 1 even counter?
3. Can we generalize this method to any number of counters with any distribution of odd/even numbers?
4. What would the probability be if we added an extra even counter?
5. How does the formula for combinations work, and why is it useful?

Tip:

When solving probability problems, always classify outcomes based on conditions (like odd/even) and use the combination formula to simplify counting.