### d) A letter from the second half of the alphabet
### Problem:
In one bag, there are five cards numbered 1, 3, 5, 7, and 9.
In another bag, there are four cards numbered 2, 4, 6, and 8.
One card is drawn at random from each bag and the values on the two cards are added.
a) Use a table to list all the possible outcomes for the activity.
b) What is the probability that:
(i) the sum is an odd number
(ii) the sum is 13
(iii) the sum is less than 10
(iv) the sum is a multiple of 5?
### Solution:
#### Part a: Listing All Possible Outcomes
To list all possible outcomes when one card is drawn from each bag and their values are summed, we create a table:
| Bag 1 \ Bag 2 | 2 | 4 | 6 | 8 |
|---------------|----|----|----|----|
| 1 | 3 | 5 | 7 | 9 |
| 3 | 5 | 7 | 9 | 11 |
| 5 | 7 | 9 | 11 | 13 |
| 7 | 9 | 11 | 13 | 15 |
| 9 | 11 | 13 | 15 | 17 |
The possible outcomes (sum values) are: 3, 5, 7, 9, 5, 7, 9, 11, 7, 9, 11, 13, 9, 11, 13, 15, 11, 13, 15, 17
We can list these outcomes as ordered pairs (bag1, bag2, sum):
(1, 2, 3), (1, 4, 5), (1, 6, 7), (1, 8, 9),
(3, 2, 5), (3, 4, 7), (3, 6, 9), (3, 8, 11),
(5, 2, 7), (5, 4, 9), (5, 6, 11), (5, 8, 13),
(7, 2, 9), (7, 4, 11), (7, 6, 13), (7, 8, 15),
(9, 2, 11), (9, 4, 13), (9, 6, 15), (9, 8, 17)
#### Part b: Calculating Probabilities
i) Probability the Sum is an Odd Number
Let's count the outcomes where the sum is an odd number: 3, 5, 7, 9, 5, 7, 9, 11, 7, 9, 11, 13, 9, 11, 13, 15, 11, 13, 15, 17. All of these sums are odd.
Thus, number of odd outcomes = 20
Total number of possible outcomes = 20
Probability the sum is odd = [tex]\(\frac{20}{20} = 1.0\)[/tex]
ii) Probability the Sum is 13
Let's count the instances where the sum is 13:
(5, 8, 13), (7, 6, 13), (9, 4, 13).
Thus, number of outcomes summing to 13 = 3
Total number of possible outcomes = 20
Probability the sum is 13 = [tex]\(\frac{3}{20} = 0.15\)[/tex]
iii) Probability the Sum is Less than 10
Let's count the instances where the sum is less than 10: 3, 5, 7, 9, 5, 7, 9, 7, 9, corresponding to the following pairs:
(1, 2, 3), (1, 4, 5), (1, 6, 7), (1, 8, 9), (3, 2, 5), (3, 4, 7), (3, 6, 9), (5, 2, 7), (5, 4, 9).
Thus, number of outcomes less than 10 = 10
Total number of possible outcomes = 20
Probability the sum is less than 10 = [tex]\(\frac{10}{20} = 0.5\)[/tex]
iv) Probability the Sum is a Multiple of 5
Let's count the instances where the sum is a multiple of 5: 5, 5, 10, corresponding to the following pairs:
(1, 4, 5), (3, 2, 5).
Thus, number of outcomes that are multiples of 5 = 4
Total number of possible outcomes = 20
Probability the sum is a multiple of 5 = [tex]\(\frac{4}{20} = 0.2\)[/tex]
