(II) Calculate the probabilities, when you throw two dice, of obtaining


(a) a 7

Verified step by step guidance

1. Identify the total number of possible outcomes when two dice are thrown. Each die has 6 faces, so the total number of outcomes is 6 faces per die multiplied by 6 faces per die.
2. Determine the favorable outcomes for obtaining a sum of 7. These outcomes are (1,6), (2,5), (3,4), (4,3), (5,2), and (6,1).
3. Count the number of favorable outcomes. From the previous step, there are 6 favorable outcomes.
4. Calculate the probability of obtaining a sum of 7 by dividing the number of favorable outcomes by the total number of outcomes.
5. Express the probability as a fraction or a decimal to complete your calculation.

Key Concepts

Probability

Probability is a measure of the likelihood that an event will occur, expressed as a number between 0 and 1. In the context of rolling dice, it quantifies how many favorable outcomes exist compared to the total possible outcomes. For example, when rolling two dice, the total number of outcomes is 36 (6 sides on the first die multiplied by 6 sides on the second).

Outcomes of Rolling Two Dice

When rolling two six-sided dice, each die can land on any number from 1 to 6, leading to a total of 36 possible combinations. To find the probability of rolling a specific sum, such as 7, we need to identify all the combinations that yield that sum. The combinations for a sum of 7 are (1,6), (2,5), (3,4), (4,3), (5,2), and (6,1), totaling 6 favorable outcomes.

Calculating Probability of a Specific Event

To calculate the probability of obtaining a specific event, such as rolling a sum of 7 with two dice, you divide the number of favorable outcomes by the total number of possible outcomes. In this case, the probability of rolling a 7 is calculated as 6 favorable outcomes divided by 36 total outcomes, resulting in a probability of 1/6 or approximately 0.167.