We are given two sets of numbers:

• Set A: $\{1, 2, 3, 4, 5, 6, 7, 8, 9\}$
• Set B: $\{4, 5, 6, 7, 8, 9\}$

We are asked to calculate the probabilities $P_1$ (the probability that the sum of a number from Set A and a number from Set B is 10) and $P_2$ (the probability that their sum is 8), then find $P_1 + P_2$.

Step 1: Total possible outcomes

Each number from Set A can be paired with each number from Set B. Since Set A contains 9 elements and Set B contains 6 elements, the total number of possible pairs is:

$9 \times 6 = 54$

Step 2: Calculating $P_1$ (Probability that the sum is 10)

We need to find all pairs where the sum of a number from Set A and a number from Set B equals 10.

Let $a \in A$ and $b \in B$. We need $a + b = 10$.

• If $a = 1$, then $b = 9$ (valid since $9 \in B$).
• If $a = 2$, then $b = 8$ (valid since $8 \in B$).
• If $a = 3$, then $b = 7$ (valid since $7 \in B$).
• If $a = 4$, then $b = 6$ (valid since $6 \in B$).
• If $a = 5$, then $b = 5$ (valid since $5 \in B$).
• If $a = 6$, then $b = 4$ (valid since $4 \in B$).

So, there are 6 pairs where the sum equals 10.

The probability $P_1$ is the number of favorable outcomes (6) divided by the total number of possible outcomes (54):

$P_1 = \frac{6}{54} = \frac{1}{9}$

Step 3: Calculating $P_2$ (Probability that the sum is 8)

Now, we find the pairs where the sum equals 8.

Let $a \in A$ and $b \in B$. We need $a + b = 8$.

• If $a = 1$, then $b = 7$ (valid since $7 \in B$).
• If $a = 2$, then $b = 6$ (valid since $6 \in B$).
• If $a = 3$, then $b = 5$ (valid since $5 \in B$).
• If $a = 4$, then $b = 4$ (valid since $4 \in B$).

So, there are 4 pairs where the sum equals 8.

The probability $P_2$ is the number of favorable outcomes (4) divided by the total number of possible outcomes (54):

$P_2 = \frac{4}{54} = \frac{2}{27}$

Step 4: Calculating $P_1 + P_2$

Now, we add $P_1$ and $P_2$:

$P_1 + P_2 = \frac{1}{9} + \frac{2}{27}$

We can write $\frac{1}{9}$ as $\frac{3}{27}$ to have a common denominator:

$P_1 + P_2 = \frac{3}{27} + \frac{2}{27} = \frac{5}{27}$

Thus, the value of $P_1 + P_2$ is $\frac{5}{27}$.

Final Answer:

$P_1 + P_2 = \frac{5}{27}$

Would you like more details, or have any questions?

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Here are 5 related questions to expand on this problem:

1. What is the probability that the sum of the two numbers is greater than 10?
2. If Set A is modified to $\{1, 2, 3, 4, 5\}$, how does $P_1 + P_2$ change?
3. What is the probability that the sum of the two numbers is less than 8?
4. Can you calculate the probability that the two numbers are equal?
5. How would you calculate the probability that the sum of the two numbers is an odd number?

Tip: When calculating probabilities, make sure to clearly define the total number of outcomes and favorable outcomes before solving.