From the set {1,2,…,15}, Alice and Bob each choose a number (different from each other). We know that Alice’s number can be divided by 5, then what is the probability that Alice’s number is larger than Bob’s?




We are given a set \( S = \{ 1, 2, \dots, 15 \} \) and told that Alice and Bob each choose a different number from this set. Alice’s number is divisible by 5, and we are asked to find the probability that Alice's number is larger than Bob’s.

### Step 1: Identify Alice's possible numbers
Since Alice’s number is divisible by 5, we first identify the numbers in the set \( S \) that are divisible by 5:
\[
\{ 5, 10, 15 \}
\]
Thus, Alice can choose one of the numbers from the set \( \{ 5, 10, 15 \} \).

### Step 2: Determine total possible outcomes
Now, let's consider how many different ways Alice and Bob can choose their numbers:
- Alice has 3 possible choices for her number (5, 10, or 15).
- Bob can choose any of the remaining 14 numbers in the set \( S \), except for the number Alice chose.

Thus, the total number of possible pairs \((A, B)\), where \(A\) is Alice's number and \(B\) is Bob's number, is:
\[
3 \times 14 = 42
\]
This is the total number of outcomes.

### Step 3: Determine favorable outcomes
We want to find the number of outcomes where Alice’s number is greater than Bob’s number. For each of Alice’s possible choices, let's count how many of Bob’s possible choices are smaller:

- **If Alice chooses 5:** Bob can only choose numbers smaller than 5, i.e., \( \{ 1, 2, 3, 4 \} \). There are 4 such choices.
- **If Alice chooses 10:** Bob can choose numbers smaller than 10, i.e., \( \{ 1, 2, 3, 4, 5, 6, 7, 8, 9 \} \). There are 9 such choices.
- **If Alice chooses 15:** Bob can choose numbers smaller than 15, i.e., \( \{ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 \} \). There are 14 such choices.

Thus, the total number of favorable outcomes is:
\[
4 + 9 + 14 = 27
\]

### Step 4: Calculate the probability
The probability that Alice’s number is larger than Bob’s number is the ratio of favorable outcomes to total outcomes:
\[
P(\text{Alice's number is larger}) = \frac{27}{42} = \frac{9}{14}
\]

Thus, the probability that Alice's number is larger than Bob's is \( \frac{9}{14} \).

Would you like further details or have any questions?

Here are 5 questions that expand on this problem:
1. How would the probability change if Alice could only choose numbers divisible by 3?
2. What is the probability that Bob’s number is larger than Alice’s number?
3. What if Alice could choose any number from the set but Bob's number had to be divisible by 3?
4. How does the total number of outcomes change if we include the condition that Bob’s number must be odd?
5. How would the problem change if Alice and Bob were choosing from the set \(\{1, 2, \dots, 20\}\)?

**Tip**: Whenever dealing with probabilities involving choosing numbers from a set, carefully count the favorable outcomes before dividing by the total outcomes.

From the set {1,2,…,15}, Alice and Bob each choose a number (different from each other). We know that Alice’s number can be divided by 5, then what is the probability that Alice’s number is larger than Bob’s?

Explore the probability of Alice’s number, divisible by 5, being larger than Bob’s number from the set {1,2,…,15}. A detailed breakdown of probability and combinatorics, including favorable and total outcomes, perfect for Grades 9-12 students.

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