Question
A poll shows that 76% of voters favor an initiative. If 7 voters are selected at random, what is the probability that exactly 4 of them will vote in favor?
Answers
GPT 3.5
To calculate the probability of exactly 4 out of 7 voters voting in favor of the initiative, we can use the binomial probability formula:
P(X = k) = (n choose k) * p^k * (1-p)^(n-k)
where:
- n is the total number of trials (7 in this case)
- k is the number of successful outcomes (4 in this case)
- p is the probability of success on a single trial (0.76 in this case)
- (n choose k) is the number of ways to choose k successes out of n trials, calculated as n! / (k! * (n-k)!)
Plugging in the values, we get:
P(X = 4) = (7 choose 4) * (0.76)^4 * (1-0.76)^(7-4)
P(X = 4) = (7 choose 4) * 0.76^4 * 0.24^3
P(X = 4) = 35 * 0.318387072 * 0.013824
P(X = 4) ≈ 35 * 0.00439615189
P(X = 4) ≈ 0.153866322
Therefore, the probability that exactly 4 out of 7 voters will vote in favor of the initiative is approximately 0.1539 or 15.39%.
P(X = k) = (n choose k) * p^k * (1-p)^(n-k)
where:
- n is the total number of trials (7 in this case)
- k is the number of successful outcomes (4 in this case)
- p is the probability of success on a single trial (0.76 in this case)
- (n choose k) is the number of ways to choose k successes out of n trials, calculated as n! / (k! * (n-k)!)
Plugging in the values, we get:
P(X = 4) = (7 choose 4) * (0.76)^4 * (1-0.76)^(7-4)
P(X = 4) = (7 choose 4) * 0.76^4 * 0.24^3
P(X = 4) = 35 * 0.318387072 * 0.013824
P(X = 4) ≈ 35 * 0.00439615189
P(X = 4) ≈ 0.153866322
Therefore, the probability that exactly 4 out of 7 voters will vote in favor of the initiative is approximately 0.1539 or 15.39%.