The unemployment rate in a city is 9%. Find the probability that at least 2 out of 8 people from this city sampled at random are unemployed.
To find the probability that at least 2 out of 8 people sampled at random are unemployed, we can use the binomial probability formula.
The binomial probability formula is given by:
P(X = k) = (n C k) * p^k * (1 - p)^(n - k)
where:
- P(X = k) is the probability of getting exactly k successes
- n is the number of trials (in this case, the number of people sampled)
- k is the number of successes (in this case, the number of unemployed people)
- p is the probability of success (in this case, the unemployment rate)
- (n C k) is the number of combinations of n items taken k at a time
In this case, we want to find the probability of getting 2, 3, 4, 5, 6, 7, or 8 unemployed people out of 8 sampled. To do this, we can calculate the individual probabilities and sum them up.
Let's break it down step by step:
Step 1: Calculate the probability of success (unemployment rate)
p = 0.09 (9% unemployment rate)
Step 2: Calculate the number of combinations (n C k)
Since we are sampling 8 people, the number of combinations is given by:
(8 C k) = 8! / (k! * (8 - k)!)
Step 3: Calculate the probability of getting k successes (unemployed people)
The probability of getting k successes is given by:
P(X = k) = (n C k) * p^k * (1 - p)^(n - k)
Step 4: Calculate the probability of getting at least 2 unemployed people
We can calculate the probability of getting 2, 3, 4, 5, 6, 7, or 8 unemployed people and sum them up:
P(X >= 2) = P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5) + P(X = 6) + P(X = 7) + P(X = 8)
Let's calculate the probability step by step:
P(X = 2) = (8 C 2) * 0.09^2 * (1 - 0.09)^(8 - 2)
P(X = 3) = (8 C 3) * 0.09^3 * (1 - 0.09)^(8 - 3)
P(X = 4) = (8 C 4) * 0.09^4 * (1 - 0.09)^(8 - 4)
P(X = 5) = (8 C 5) * 0.09^5 * (1 - 0.09)^(8 - 5)
P(X = 6) = (8 C 6) * 0.09^6 * (1 - 0.09)^(8 - 6)
P(X = 7) = (8 C 7) * 0.09^7 * (1 - 0.09)^(8 - 7)
P(X = 8) = (8 C 8) * 0.09^8 * (1 - 0.09)^(8 - 8)
Finally, sum up the individual probabilities to get the probability of at least 2 unemployed people:
P(X >= 2) = P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5) + P(X = 6) + P(X = 7) + P(X = 8)
Calculating each individual probability and summing them up will give you the final answer.