To solve these probability and odds questions, we need to understand the outcomes of rolling two dice.

1. Probability of Rolling a 7

When two six-sided dice are rolled, each die has 6 faces, resulting in a total of $6 \times 6 = 36$ possible outcomes. We need to determine how many of these outcomes sum to 7.

The pairs of numbers on two dice that add up to 7 are:

• (1, 6)
• (2, 5)
• (3, 4)
• (4, 3)
• (5, 2)
• (6, 1)

There are 6 pairs that add up to 7. Therefore, the probability $P$ of rolling a 7 is the number of favorable outcomes divided by the total number of possible outcomes.

$P(\text{rolling a 7}) = \frac{\text{Number of favorable outcomes}}{\text{Total possible outcomes}} = \frac{6}{36} = \frac{1}{6}$

So, the probability of rolling a 7 is $\frac{1}{6}$.

2. Odds of Rolling a 7

The odds of an event are typically expressed as the ratio of the number of ways the event can occur to the number of ways the event cannot occur.

From the probability calculation, we know:

• There are 6 ways to roll a 7.
• There are 36 total possible outcomes.
• Therefore, there are $36 - 6 = 30$ ways to roll something other than a 7.

The odds $O$ of rolling a 7 are given by:

$O(\text{rolling a 7}) = \frac{\text{Number of favorable outcomes}}{\text{Number of unfavorable outcomes}} = \frac{6}{30} = \frac{1}{5}$

So, the odds of rolling a 7 are 1 to 5.

Summary:

1. The probability of rolling a 7 is $\frac{1}{6}$.
2. The odds of rolling a 7 are 1 to 5.

Would you like any further details or have any questions?

Here are 5 numbered questions you might want to ask next:

1. What is the probability of rolling a sum of 8 with two dice?
2. What are the odds of rolling a sum less than 5 with two dice?
3. How do you calculate the expected value of a single die roll?
4. What is the probability of rolling doubles with two dice?
5. How do you calculate the probability of rolling an even number with a single die?

Tip: When dealing with dice, it's helpful to list all possible outcomes to understand the total number of favorable and unfavorable outcomes.