1. To calculate the probabilities of the coin flip outcomes, you need to determine the total number of possible outcomes and the number of favorable outcomes for each scenario.
a) Getting all tails: When flipping a fair coin, each flip has two possible outcomes, either heads or tails. Since you are flipping the coin 10 times, the total number of possible outcomes can be calculated as 2 raised to the power of 10, which is 1024.
The number of favorable outcomes for getting all tails is 1, as there is only one way to get all tails. Therefore, the probability of getting all tails is 1/1024.
b) Getting all heads: Similar to the previous scenario, the total number of possible outcomes is still 1024. However, in this case, the number of favorable outcomes is also 1 since there is only one way to get all heads. Thus, the probability of getting all heads is also 1/1024.
c) Getting at least one tails: To calculate the probability of getting at least one tails, you need to find the complement of getting all heads. The complement rule states that the probability of an event occurring is equal to 1 minus the probability of its complement.
In this case, the complement of getting all heads is getting at least one tails. So the probability of getting at least one tails can be calculated as 1 - (1/1024), which simplifies to 1023/1024.
2. When rolling a pair of fair dice, the probabilities of the different sums can be determined by considering the total number of possible outcomes and the favorable outcomes for each scenario.
a) Getting a sum of 1: The minimum sum you can get on the two dice is 2 (1 on each die), and the maximum sum is 12 (6 on each die). Since the sum of 1 is not possible, the probability of getting a sum of 1 is 0.
b) Getting a sum of 5: To calculate the probability of getting a sum of 5, you need to determine how many unique outcomes result in a sum of 5. In this case, there are four possible combinations: (1, 4), (2, 3), (3, 2), and (4, 1). Since there are 36 total possible outcomes when rolling a pair of dice (6 outcomes on the first die multiplied by 6 outcomes on the second die), the probability of getting a sum of 5 is 4/36, which simplifies to 1/9.
c) Getting a sum of 12: There is only one way to get a sum of 12, which is by rolling two 6's. Since there are still 36 total possible outcomes, the probability of getting a sum of 12 is 1/36.