M is an integer randomly chosen from the set $ \{7, 14, 25, 27, 33, 29, 30\} $.

**Quantity I:** Probability that the average of 12, 9, and M is at least 17.

**Quantity II:** $ \frac{1}{3} $

Compare Quantity I and Quantity II and choose the correct option.

Question

M is an integer randomly chosen from the set $ \{7, 14, 25, 27, 33, 29, 30\} $.

**Quantity I:** Probability that the average of 12, 9, and M is at least 17.

**Quantity II:** $ \frac{1}{3} $

Compare Quantity I and Quantity II and choose the correct option.

Accepted Answer

**Solution:**

1. The average of 12, 9, and M is given by:
   $$\frac{12 + 9 + M}{3}$$

2. We want the probability that this average is at least 17, so:
   $$\frac{21 + M}{3} \geq 17$$

3. Multiply both sides by 3:
   $$21 + M \geq 51$$

4. Subtract 21 from both sides:
   $$M \geq 30$$

5. From the set $ \{7, 14, 25, 27, 33, 29, 30\} $, the values satisfying $ M \geq 30 $ are 30 and 33.

6. Number of favorable outcomes = 2
   Total number of outcomes = 7

7. Probability (Quantity I) = $ \frac{2}{7} \approx 0.2857 $

8. Quantity II = $ \frac{1}{3} \approx 0.3333 $

Since $ \frac{2}{7} < \frac{1}{3} $, Quantity I is less than Quantity II.

**Answer:** Quantity I $ < $ Quantity II

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