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A local county has an unemployment rate of 5.1%. A random sample of 6 employable people are picked at random from the county and are asked if they are employed. The distribution is a binomial. Round answers to 4 decimal places. a) Find the probability that exactly 2 in the sample are unemployed.

User Iain Rist
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To calculate the probability of exactly 2 people out of a sample of 6 being unemployed, we use the Binomial Probability formula:

P(x=k) = C(n, k) * (p^k) * ((1-p)^(n-k))

Here,

- n is the number of trials or the sample size, which is 6 in our case.
- p is the probability of the event in question, the unemployment rate in our case, which is 5.1%, or 0.051.
- k is the number of times the event is to occur, which is finding exactly 2 unemployed people in this case.

Now, let's plug the values into the formula:

1) Calculate the binomial coefficient C(n, k) (also known as "n choose k"), which is the number of combinations of n things taken k at a time. In this case, it's "6 choose 2".

2) Calculate p^k, which is (0.051)^2.

3) Calculate (1-p)^(n-k), which is (1-0.051)^(6-2).

Finally, multiply these three values together to get the probability.

After the necessary calculations, the probability that exactly 2 out of the 6 employable people from the sample are unemployed would be approximately 0.0316, or 3.16% when rounded to 4 decimal places. This implies that if you sample six people from this population, there's a 3.16% chance that exactly two of them will be unemployed. Remember that this result is specific to the conditions given and might change as those parameters change.

User Ednincer
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