Teenage smoking rate dropped to 8.5% in 2007 from 17.6% in 2001 and 23% in 1997. 1. Calculate the probability that at least one in a group of 10 teenagers smoked in 2007. 2. Calculate the probability that at least one in a group of 10 teenagers smoked in 2001. 3. Calculate the probability that at least one in a group of 10 teenagers smoked in 1997.

To calculate the probabilities, we need to first find the probability that a single teenager did not smoke in each year.

1. In 2007, the smoking rate was 8.5%. This means that the probability of a single teenager not smoking is 1 - 0.085 = 0.915.

2. In 2001, the smoking rate was 17.6%. This means that the probability of a single teenager not smoking is 1 - 0.176 = 0.824.

3. In 1997, the smoking rate was 23%. This means that the probability of a single teenager not smoking is 1 - 0.23 = 0.77.

Next, we can calculate the probability that at least one teenager in a group of 10 smoked. We can use the complement rule, which states that the probability of an event occurring is equal to one minus the probability of the event not occurring.

For example, to calculate the probability that at least one in a group of 10 teenagers smoked in 2007:

P(at least one teenager smoked in 2007) = 1 - P(no teenager smoked in 2007)

P(no teenager smoked in 2007) = (probability of a single teenager not smoking in 2007) ^ number of teenagers

P(no teenager smoked in 2007) = 0.915^10 ≈ 0.302

P(at least one teenager smoked in 2007) = 1 - 0.302 = 0.698 (approximately)

Similarly, you can calculate the probabilities for 2001 and 1997 by substituting the respective values. Use the same procedure for each year:

1. P(at least one teenager smoked in 2001) = 1 - P(no teenager smoked in 2001)
P(no teenager smoked in 2001) = (0.824)^10 ≈ 0.215

2. P(at least one teenager smoked in 1997) = 1 - P(no teenager smoked in 1997)
P(no teenager smoked in 1997) = (0.77)^10 ≈ 0.05

Therefore:

1. The probability that at least one in a group of 10 teenagers smoked in 2007 is approximately 0.698.
2. The probability that at least one in a group of 10 teenagers smoked in 2001 is approximately 0.215.
3. The probability that at least one in a group of 10 teenagers smoked in 1997 is approximately 0.05.