Among 26-31 year old, 29% say they have written an editorial letter while under the influence of peer pressure. Suppose six 26-31 year old are selected at random. (a) that at least one has not written under the influence of peer pressure. (b) what is the probability of that at least one has written an editorial letter while under the influence of peer pressure.

is part (a) a given condition or you're expected to calculate the said event?

it is a given

If at least one has not written, then there are only 5 to consider.

p=0.29
q=(1-0.29)=0.71

P(at least one out of 5)
=1-P(none out of 5)
=1-q^5
=1-0.18
=0.82

To find the probability in both scenarios, we can use the concept of complementary events. The complementary event of an event A is the event that A does not occur. In this case, event A is selecting at least one person who has not written under the influence of peer pressure, and event B is selecting at least one person who has written an editorial letter under the influence of peer pressure.

Let's calculate the probabilities step by step:

(a) Probability that at least one person has not written under the influence of peer pressure:
To find the probability of at least one person not having written under the influence of peer pressure, we can use the complementary event. The probability that all six selected people have written under the influence of peer pressure is 29% raised to the power of 6 (as the probability of each person writing under the influence is 29%). Thus, the complementary event is 1 - P(all six have written under the influence of peer pressure):

P(at least one has not written under the influence of peer pressure) = 1 - P(all six have written under the influence of peer pressure)
P(at least one has not written under the influence of peer pressure) = 1 - (0.29)^6
P(at least one has not written under the influence of peer pressure) ≈ 1 - 0.006025905
P(at least one has not written under the influence of peer pressure) ≈ 0.993974095

Therefore, the probability that at least one person has not written under the influence of peer pressure is approximately 0.993974095, or 99.397%.

(b) Probability that at least one person has written an editorial letter while under the influence of peer pressure:
We can use the same approach as above, but this time we want to find the probability that at least one person has written under the influence of peer pressure. The probability of the complementary event (none of the six people writing under the influence) is (1 - 0.29)^6:

P(at least one has written under the influence of peer pressure) = 1 - P(none have written under the influence of peer pressure)
P(at least one has written under the influence of peer pressure) = 1 - (1 - 0.29)^6
P(at least one has written under the influence of peer pressure) ≈ 1 - 0.641323759
P(at least one has written under the influence of peer pressure) ≈ 0.358676241

Therefore, the probability that at least one person has written an editorial letter under the influence of peer pressure is approximately 0.358676241, or 35.868%.