In this exercise, we take a look at 12 common probability and dice questions, and their corresponding answers related to **two dice are thrown simultaneously** . From calculating the likelihood of specific outcomes when throwing two dice to analysing the probabilities of obtaining certain sums or number patterns, these questions provide a comprehensive overview of concepts related to dice rolls.

**Question 1: When two dice are thrown, what is the probability of getting a sum of seven?**

**Answer:**

The possible combinations that result in a sum of seven when two dice are thrown are (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), and (6, 1). There are six possible outcomes out of a total of 36 outcomes, so the probability is 6/36 = 1/6.

**Question 2: When two dice are thrown, what is the probability of getting an even number on both dice?**

**Answer:**

There are 15 possible outcomes that result in an even number on both dice, namely: (2, 2), (2, 4), (2, 6), (4, 2), (4, 4), (4, 6), (6, 2), (6, 4), (6, 6), (3, 5), (5, 3), (3, 1), (1, 3), (5, 1), and (1, 5). There are a total of 36 possible outcomes when two dice are thrown. So, the probability of getting an even number on both dice is 15/36 = 5/12.

### Question 3: When two dice are thrown, what is the probability of getting a sum that is a prime number?

**Answer:**

The prime numbers that can be obtained as sums when two dice are thrown are 2, 3, 5, 7, 11. There are 15 possible outcomes (36 total outcomes) that give a sum of a prime number. So, the probability is 15/36 = 5/12.

### Question 4: Two dice are thrown simultaneously, what is the probability of getting two numbers and their product is even?

**Answer:**

To obtain two numbers with an even product when two dice are thrown simultaneously, the following outcomes are favourable: (2, 2), (2, 4), (2, 6), (4, 2), (4, 4), (4, 6), (6, 2), (6, 4), and (6, 6). There are nine favourable outcomes out of a total of 36 outcomes, so the probability is 9/36 = 1/4.

**Question 5: Two dice are thrown simultaneously. If X denotes the number of sixes, find the expectation of X?**

**Answer:**

The expectation of X, denoting the number of sixes when two dice are thrown simultaneously, can be calculated using the formula for the expectation of a discrete random variable. Since the probability of getting a six on a single die is 1/6, the expectation of X is 2 * (1/6) = 1/3.

### Question 6: If you roll two fair six-sided dice, what is the probability that at least one die shows a 3?

**Answer:**

The probability that at least one die shows a 3 when two fair six-sided dice are rolled can be found by calculating the complement of the probability that no die shows a 3. The probability of not getting a 3 on a single die is 5/6, so the probability of not getting a 3 on both dice is (5/6) * (5/6) = 25/36. Therefore, the probability of at least one die showing a 3 is 1 – 25/36 = 11/36.

### Question 7: Two dice are rolled what is the probability of not getting doubles?

**Answer:**

The probability of not getting doubles when two dice are rolled can be found by calculating the complement of the probability of getting doubles. There are six possible outcomes that result in doubles (e.g., (1, 1), (2, 2), etc.) out of a total of 36 possible outcomes. So, the probability of getting doubles is 6/36 = 1/6. Therefore, the probability of not getting doubles is 1 – 1/6 = 5/6.

### Question 8: If you roll two fair six-sided dice, what is the probability that the sum is 9.99 or higher?

**Answer:**

The probability of getting a sum of 9.99 or higher when two fair six-sided dice are rolled is 0, since the highest possible sum is 12.

**Question 9: Two dice are thrown simultaneously. What is the probability that the sum is greater than or equal to 7?**

**Answer:**

There are 15 possible outcomes that result in a sum greater than or equal to 7 when two dice are thrown simultaneously, namely: (3, 4), (3, 5), (3, 6), (4, 3), (4, 4), (4, 5), (4, 6), (5, 3), (5, 4), (5, 5), (5, 6), (6, 3), (6, 4), (6, 5), and (6, 6). There are a total of 36 possible outcomes when two dice are thrown. So, the probability of getting a sum greater than or equal to 7 is 15/36 = 5/12.

### Question 10: Two dice are thrown simultaneously. What is the probability that the sum is less than or equal to 6?

**Answer:**

There are 21 possible outcomes that result in a sum less than or equal to 6 when two dice are thrown simultaneously. So, the probability of getting a sum less than or equal to 6 is 21/36 = 7/12.

### Question 11: Two dice are thrown simultaneously. What is the probability that the sum is equal to 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, or 12?

**Answer:**

There are 36 possible outcomes when two dice are thrown simultaneously. So, the probability of getting any sum between 2 and 12 is 1.

### Question 12: Two dice are thrown simultaneously. What is the probability that the sum is not equal to 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, or 12?

**Answer:**

Since the probability of getting any sum between 2 and 12 is 1, the probability of getting a sum that is not equal to any of those numbers is 0.

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