A tetrahedron has four congruent triangular faces labeled 10, 20, 30, and 40. Let T represent "the tetrahedron was rolled." Let S represent the event "lands on a multiple of 7." Find P(S|T). 

We are given that a tetrahedron has four congruent triangular faces labeled 10, 20, 30, and 40. The event \( T \) represents "the tetrahedron was rolled," and the event \( S \) represents "lands on a multiple of 7." We are tasked with finding \( P(S \mid T) \), the probability that the tetrahedron lands on a multiple of 7 given that it was rolled.

### Step 1: Determine the sample space
The tetrahedron has four faces labeled 10, 20, 30, and 40. These are the possible outcomes when the tetrahedron is rolled. Thus, the sample space \( \Omega \) is:

\[
\Omega = \{ 10, 20, 30, 40 \}
\]

### Step 2: Identify multiples of 7
The multiples of 7 within the sample space are the values that satisfy \( x \equiv 0 \pmod{7} \). We check each face value:

- \( 10 \div 7 \) gives a remainder of 3, so 10 is not a multiple of 7.
- \( 20 \div 7 \) gives a remainder of 6, so 20 is not a multiple of 7.
- \( 30 \div 7 \) gives a remainder of 2, so 30 is not a multiple of 7.
- \( 40 \div 7 \) gives a remainder of 5, so 40 is not a multiple of 7.

Thus, there are no multiples of 7 in the set \( \{ 10, 20, 30, 40 \} \).

### Step 3: Calculate \( P(S \mid T) \)
Since there are no multiples of 7 in the sample space, the event \( S \) cannot occur. Therefore, the probability of landing on a multiple of 7 given that the tetrahedron was rolled is:

\[
P(S \mid T) = 0
\]

Thus, \( P(S \mid T) = 0 \).

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Do you need any further details or have any questions?

Here are some related questions for further exploration:
1. What is the probability of landing on a number greater than 25?
2. How would the probability change if the face values were multiples of 5?
3. What if one of the faces were labeled with a multiple of 7—how would that affect \( P(S \mid T) \)?
4. How is the probability affected if the tetrahedron is weighted towards certain faces?
5. What if the tetrahedron had 6 faces, each labeled with different numbers?

**Tip**: When calculating conditional probabilities, always verify whether the event in question can actually occur under the given conditions.

A tetrahedron has four congruent triangular faces labeled 10, 20, 30, and 40. Let T represent 'the tetrahedron was rolled.' Let S represent the event 'lands on a multiple of 7.' Find P(S|T).

Explore how to find the probability of rolling a tetrahedron with faces labeled 10, 20, 30, and 40, and landing on a multiple of 7. This problem involves the concepts of probability and conditional probability, suitable for students in Grades 9-12.

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