To determine the theoretical probabilities, we'll analyze the different possible outcomes for each case.
Total number of cards = 40 (5 cards for each number from 2 to 9)
a) Sum is 12:
There are several combinations of cards that can result in a sum of 12:
- 3 and 9
- 4 and 8
- 5 and 7
- 7 and 5
- 8 and 4
- 9 and 3
Total favorable outcomes = 6 (since there are 6 possible combinations)
Total possible outcomes (since you're drawing two cards) = 40 * 40 = 1600
Probability (sum is 12) = (Number of favorable outcomes) / (Total possible outcomes)
Probability (sum is 12) = 6 / 1600 = 0.00375
b) Difference is less than 5:
There are several combinations of cards that can result in a difference less than 5:
- 2 and 3
- 3 and 2
- 3 and 4
- 4 and 3
- 4 and 5
- 5 and 4
- 5 and 6
- 6 and 5
- 6 and 7
- 7 and 6
- 7 and 8
- 8 and 7
- 8 and 9
- 9 and 8
Total favorable outcomes = 14
Total possible outcomes = 1600
Probability (difference is less than 5) = 14 / 1600 = 0.00875
c) Product is odd:
For the product to be odd, one or both of the numbers drawn must be odd. Since there are five odd numbers (3, 5, 7, 7, 9), and each number has five cards, the total number of favorable outcomes is 5 + 5 + 5 + 5 + 5 = 25.
Total possible outcomes = 1600
Probability (product is odd) = 25 / 1600 = 0.015625
To summarize:
- a) Probability (sum is 12) ≈ 0.00375
- b) Probability (difference is less than 5) ≈ 0.00875
- c) Probability (product is odd) ≈ 0.015625