A triangular die is a four-sided die, each side possessing either a number 1, 2, 3, or 4. Two such dice are tossed simultaneously and the bottom faces - the face that each die lands on - is observed. Assume the two dice are fair. Consider the following events:

the bottom-most faces sum to six

each bottom-most face shows an even number

both bottom-most faces show the same number, referred to as

Part (a) What is the probability that the bottom-most faces do not sum to six?
use event a and b but from be only use 2,4 4,2 because those sum to 6

Let's analyze the problem by defining the events clearly and calculating the probabilities based on the information provided.

### Part (a) Probability that the bottom-most faces do not sum to six

Given events:

1. **Event A**: The bottom-most faces sum to six.
2. **Event B**: Each bottom-most face shows an even number (possible pairs are (2,2), (2,4), (4,2), (4,4)).

However, you mentioned using only (2,4) and (4,2) from Event B because those sum to 6.

**Possible outcomes when two triangular dice are rolled:**

Each die has four faces, numbered 1, 2, 3, and 4. The total number of possible outcomes when two dice are rolled is:

\[
4 \times 4 = 16
\]

**Event A: Pairs that sum to 6**

The pairs of numbers that sum to 6 are:

- (2, 4)
- (4, 2)
- (3, 3)

So, there are 3 outcomes that satisfy Event A.

**Probability of Event A (sum to six):**

\[
P(A) = \frac{\text{Number of outcomes summing to 6}}{\text{Total outcomes}} = \frac{3}{16}
\]

**Probability of the complement of Event A (do not sum to 6):**

The probability that the bottom-most faces do not sum to six is the complement of Event A:

\[
P(\text{not } A) = 1 - P(A) = 1 - \frac{3}{16} = \frac{13}{16}
\]

### Final Answer:

The probability that the bottom-most faces do not sum to six is:

\[
\boxed{\frac{13}{16}}
\]

Would you like further details or have any questions? Here are some related questions that could expand on this information:

1. What is the probability that the sum of the faces is greater than 5?
2. What is the probability that both bottom-most faces show the same number?
3. What is the probability that at least one die shows an odd number?
4. What is the probability that the sum of the bottom-most faces is an odd number?
5. How would the probabilities change if the dice were not fair?

**Tip:** Remember, when calculating probabilities of events, the complement rule (i.e., \( P(\text{not A}) = 1 - P(A) \)) can simplify finding probabilities of outcomes not directly listed.

A triangular die is a four-sided die, each side possessing either a number 1, 2, 3, or 4. Two such dice are tossed simultaneously and the bottom faces - the face that each die lands on - is observed. Assume the two dice are fair. Consider the following events:

the bottom-most faces sum to six
each bottom-most face shows an even number
both bottom-most faces show the same number
Part (a) What is the probability that the bottom-most faces do not sum to six?

Calculate the probability that two four-sided dice do not sum to six. This problem covers key concepts in probability, including event complements and combinatorics. Learn how to use the complement rule to simplify calculations. Suitable for high school students (Grades 9-12).

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