Source: ballotpedia.org/Presidential_election_in_Washington,_D.C.,_2020
Question: If 5.4% of voters from a certain city are Republicans, what is the probability that a randomly picked twelve-person jury of people who live in that city will have one Republican?
AI answer: To calculate the probability of randomly selecting a twelve-person jury from a certain city and having one Republican, we need to use the concept of binomial probability. Binomial probability is used to determine the likelihood of a specific number of successes (in this case, having one Republican) in a fixed number of independent trials (selecting twelve people).
The probability of selecting one Republican out of twelve people can be calculated using the binomial probability formula:
P(X = k) = C(n, k) * p^k * (1-p)^(n-k)
Where:
– P(X = k) is the probability of getting exactly k successes
– C(n, k) is the number of combinations of n items taken k at a time
– p is the probability of success on any given trial
– n is the total number of trials
In this case, we have n = 12 (as we are selecting twelve people), and p = 0.054 (as 5.4% of voters are Republicans). We want to find P(X = 1), which represents the probability of having one Republican in the jury.
Using the formula, we can calculate:
P(X = 1) = C(12, 1) * (0.054)^1 * (1-0.054)^(12-1)
C(12, 1) represents the number of ways to choose one Republican out of twelve people, which can be calculated as:
C(12, 1) = 12! / (1! * (12-1)!) = 12
Substituting these values into the formula:
P(X = 1) = 12 * (0.054)^1 * (0.946)^11
Calculating this expression gives us the probability that a randomly picked twelve-person jury will have one Republican from that city.
P(X = 1) = 12 * (0.054)^1 * (0.946)^11 ≈ 0.283
Therefore, the probability that a randomly picked twelve-person jury of people who live in that city will have one Republican is approximately 0.283, or 28.3%.
This is the correct question:
If 5.4% of voters from a certain city are Republicans, what is the probability that a randomly picked twelve-person jury of people who live in that city will have one or more Republicans?
The probability that a randomly picked jury of 12 people will have no Republicans is:
(0.946)^12 = 0.561004829523521
So, the probability that the jury will have one or more Republicans is 1 minus this probability, or:
1 – 0.561004829523521 = 0.438995170476479
Therefore, the probability that a randomly picked jury of 12 people from this city will have one or more Republicans is 43.8995170476479%.
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