Suppose that about 25% of those called will find and excuse (work, poor health, etc) to avoid jury duty. If 12 people are called for jury duty, what is the probablity that 4 or more will not be available to serve on the jury?
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To find the probability that 4 or more people will not be available to serve on the jury, we need to calculate the probability that exactly 4, exactly 5, all the way up to exactly 12 people will not be available, and then add up those probabilities.
First, let's calculate the probability that exactly 4 people will not be available. The probability that a person will not be available is 25%, or 0.25. So, the probability that exactly 4 people will not be available is calculated as:
P(4) = C(12, 4) * (0.25)^4 * (0.75)^(12-4)
Here, C(12, 4) represents the binomial coefficient, which calculates the number of ways to choose 4 people from a group of 12. It can be calculated as:
C(12, 4) = 12! / (4! * (12-4)!)
Using factorials, we can simplify:
C(12, 4) = (12 * 11 * 10 * 9) / (4 * 3 * 2 * 1) = 495
Now, let's calculate the probability for exactly 5, 6, 7, 8, 9, 10, 11, and 12 people not being available. We will use the same formula for each case, substituting the appropriate values:
P(5) = C(12, 5) * (0.25)^5 * (0.75)^(12-5)
P(6) = C(12, 6) * (0.25)^6 * (0.75)^(12-6)
P(7) = C(12, 7) * (0.25)^7 * (0.75)^(12-7)
P(8) = C(12, 8) * (0.25)^8 * (0.75)^(12-8)
P(9) = C(12, 9) * (0.25)^9 * (0.75)^(12-9)
P(10) = C(12, 10) * (0.25)^10 * (0.75)^(12-10)
P(11) = C(12, 11) * (0.25)^11 * (0.75)^(12-11)
P(12) = C(12, 12) * (0.25)^12 * (0.75)^(12-12)
Finally, to get the probability that 4 or more people will not be available, we add up these probabilities:
P(4 or more) = P(4) + P(5) + P(6) + P(7) + P(8) + P(9) + P(10) + P(11) + P(12)
By plugging in the values and calculating each probability, you will be able to find the answer to the question.