Question

among 51 - to 56 - year - olds, 29% say they have written an editorial letter while under the influence of alcohol. suppose seven 51 - to 56 - year - olds are selected at random. complete parts (a) through (d) below. (a) what is the probability that all seven have written an editorial letter while under the influence of alcohol? (round to four decimal places as needed.)

Explanation:

Step1: Identify the probability of a single - person event

The probability that a 51 - to 56 - year - old has written an editorial letter while under the influence of alcohol is $p = 0.29$.

Step2: Use the multiplication rule for independent events

Since the selection of each person is independent, the probability that all $n = 7$ people have written an editorial letter while under the influence of alcohol is given by the formula $P(X = 7)=p^n$. Substitute $p = 0.29$ and $n = 7$ into the formula: $P(X = 7)=(0.29)^7$.

Step3: Calculate the result

$(0.29)^7=0.29\times0.29\times0.29\times0.29\times0.29\times0.29\times0.29\approx0.0001$.

Answer:

$0.0001$