From a lot of 12 flares, 4 are selected at random. if the lot has 4 defective flares, what is the probability that all 4 will not work.
B. whats the pobablity that at most 2 will not work?
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To find the probability in both cases, we need to first determine the total number of possible outcomes and the number of favorable outcomes.
1. Probability that all 4 flares will not work:
Since there are 12 flares in total and 4 are selected randomly, the total number of possible outcomes can be calculated using combinations. This can be written as C(12, 4), which represents choosing 4 items from a set of 12 without replacement. The formula for combinations, nCr, is n! / [(n-r)! * r!], where n is the total number of items and r is the number of items selected.
Using the formula, we can calculate:
C(12, 4) = 12! / [(12-4)! * 4!]
= 12! / (8! * 4!)
= (12 * 11 * 10 * 9) / (4 * 3 * 2 * 1)
= 495
Out of the 12 flares, 4 are defective. So, in order to select 4 flares that will not work, we need to choose them from the 8 non-defective flares. This can be calculated using combinations as well:
C(8, 4) = 8! / [(8-4)! * 4!]
= 8! / (4! * 4!)
= (8 * 7 * 6 * 5) / (4 * 3 * 2 * 1)
= 70
Therefore, the probability that all 4 flares will not work is:
Probability = favorable outcomes / total outcomes
= 70 / 495
≈ 0.1414 or 14.14%
2. Probability that at most 2 flares will not work:
To find this probability, we need to consider all the possible outcomes where 0, 1, or 2 flares do not work.
- For 0 flares not working:
We choose all 4 flares from the 8 non-defective ones, which can be calculated as C(8, 4).
Probability = C(8, 4) / C(12, 4)
- For 1 flare not working:
We choose 1 defective flare (out of the 4) to not work and the remaining 3 flares from the 8 non-defective ones.
Probability = (C(4, 1) * C(8, 3)) / C(12, 4)
- For 2 flares not working:
We choose 2 defective flares (out of the 4) to not work and the remaining 2 flares from the 8 non-defective ones.
Probability = (C(4, 2) * C(8, 2)) / C(12, 4)
To find the probability that at most 2 flares will not work, we add up these probabilities:
Probability = (C(8, 4) + (C(4, 1) * C(8, 3)) + (C(4, 2) * C(8, 2))) / C(12, 4)
Now, you can use these formulas to calculate the probabilities in both cases.