## To find the probability of getting a specific outcome when multiple events are happening, you can multiply the individual probabilities together. Let's go through each question and break down how to calculate the probabilities:

1. Probability of getting heads and a five:

The probability of getting heads when tossing a fair coin is 0.5 (or 50%). The probability of rolling a five on a fair six-sided die is 1/6, which is approximately 0.17. To find the probability of both events happening, you multiply the probabilities together: 0.5 * 0.17 = 0.085, or approximately 0.085.

2. Probability that all five polls are accurate with a margin of error:

If each poll has a confidence level of 95%, it means there is a 95% chance of the poll accurately representing the true result within a given margin of error. The probability that all five polls are accurate can be calculated by multiplying the individual probabilities together: 0.95 * 0.95 * 0.95 * 0.95 * 0.95 â‰ˆ 0.774, or approximately 77.4%.

3. Probability of selecting all Republicans from the Senate committee:

In the 105th Congress, there are 55 Republicans and 45 Democrats in the Senate. When randomly selecting 4 out of the total 100 senators, the probability of selecting all Republicans can be calculated as follows: (55/100) * (54/99) * (53/98) * (52/97) â‰ˆ 0.073, or approximately 7.3%.

4. Probability of exactly 10 flights arriving on time:

If Southwest Airlines has an 80% on-time rate, the probability of a single flight arriving on time is 0.8. To calculate the probability that exactly 10 out of 15 flights arrive on time, you use the binomial probability formula: P(X = k) = (n choose k) * p^k * (1 - p)^(n-k), where n is the number of trials, p is the probability of success, and k is the number of successful trials. Plugging in the values, the calculation would look like: (15 choose 10) * (0.8)^10 * (0.2)^(15-10).

5. Probability of at least 10 flights arriving on time:

To find the probability of at least 10 flights arriving on time, you need to calculate the probability of 10, 11, 12, 13, 14, and 15 flights arriving on time, and then add them together. You can use the formula P(X â‰¥ k) = Î£ P(X = i) for i = k to n, where Î£ denotes the summation symbol. Using this formula, you would calculate the individual probabilities for 10, 11, 12, 13, 14, and 15 flights arriving on time and sum them up.

6. Probability of at least 10 flights arriving late:

To find the probability of at least 10 flights arriving late, you can subtract the probability of at least 10 flights arriving on time from 1, because either the flights arrive on time or they arrive late. So P(X â‰¥ 10 flights arrive late) = 1 - P(X â‰¥ 10 flights arrive on time).

Regarding the question "Would it be unusual for Southwest to have 5 flights late," it would depend on the probability or percentage of flights that typically arrive late for Southwest Airlines. If the percentage of late flights is significantly lower than 20% (the complement of the on-time rate of 80%), then it would be considered unusual for them to have 5 flights late.