35% of North Americans do not think that college education is a contributory factor to economic success. A random sample of seven North Americans is selected. Find the probability that:
a. Exactly 3 people will agree with that position (taken by the 35% of the population)
b. At least 3 people will agree with the position
c. At most 5 people will agree with the position
d. Fewer than 4 people will agree with the position
I did this one earlier today
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To find the probability for each of these scenarios, we can use the binomial probability formula:
P(X = k) = C(n, k) * p^k * (1 - p)^(n - k)
Where:
P(X = k) is the probability of 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 for each trial,
n is the number of trials,
k is the number of successes.
Given information:
p = 0.35 (probability of agreeing with the position)
n = 7 (number of people in the sample)
a. Exactly 3 people will agree with that position (taken by the 35% of the population):
P(X = 3) = C(7, 3) * (0.35)^3 * (1 - 0.35)^(7 - 3)
To calculate C(7, 3), we can use the combination formula:
C(7, 3) = 7! / (3! * (7 - 3)!) = 35
Plug in the values:
P(X = 3) = 35 * (0.35)^3 * (0.65)^4
P(X = 3) ≈ 0.2679 (approximately)
b. At least 3 people will agree with the position:
P(X ≥ 3) = P(X = 3) + P(X = 4) + P(X = 5) + P(X = 6) + P(X = 7)
Using the same formula as above, you can calculate the probabilities for each individual case and sum them up.
c. At most 5 people will agree with the position:
P(X ≤ 5) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5)
Using the same formula as above, you can calculate the probabilities for each individual case and sum them up.
d. Fewer than 4 people will agree with the position:
P(X < 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)
Using the same formula as above, you can calculate the probabilities for each individual case and sum them up.
Remember to calculate each individual probability by plugging the values into the formula and then summing them up for the cumulative probabilities.