位置4的问题 A particular illness has a 30% recovery rate within 2 weeks. If Doctors observe a cohort of 10 patients who have that illness for two weeks, what is the probability that at least 6 of those patients will recover in that time? (Please record your answer as a probability value between zero and 1 to three digits after the decimal). 答案A particular illness has a 30% recovery rate within 2 weeks. If Doctors observe a cohort of 10 patients who have that illness for two weeks, what is the probability that at least 6 of those patients will recover in that time? (Please record your answer as a probability value between zero and 1 to three digits after the decimal). [input]

Question

BlackTom AI · Accepted Answer

We start by identifying the scenario: we have a binomial distribution with n = 10 trials (patients) and a success probability p = 0.3 (recovery within two weeks). We are asked for the probability that at least 6 patients recover, i.e., P(X ≥ 6).

To compute this, sum the probabilities for k = 6, 7, 8, 9, 10 recoveries:
P(X ≥ 6) = Σ_{k=6}^{10} C(10, k) (0.3)^k (0.7)^{10−k}.

Compute each term step by step:
- For k = 6: C(10,6) = 210. (0.3)^6 ≈ 0.000729. (0.7)^4 ≈ 0.2401. Product ≈ 0.000729 × 0.2401 ≈ 0.000175. Multiply by 210 → ≈ 0.03675.
- For k = 7: C(10,7) = 120. (0.3)^7 ≈ 0.0002187. (0.7)^3 ≈ 0.343. Product ≈ 0.0002187 × 0.343 ≈ 0.000073. Multiply by 120 → ≈ 0.0090.
- For k = 8: C(10,8) = 45. (0.3)^8 ≈ 0.00006561. (0.7)^2 = 0.49. Product ≈ 0.00006561 × 0.49 ≈ 0.000032. Multiply by 45 → ≈ 0.00145.
- For k = 9: C(10,9) = 10. (0.3)^9 ≈ 0.000019683. (0.7)^1 = 0.7. Product ≈ 0.000019683 × 0.7 ≈ 0.00001378. Multiply by 10 → ≈ 0.000138.
- For k = 10: C(10,10) = 1. (0.3)^{10} ≈ 0.000005905. (0.7)^0 = 1. Product ≈ 0.000005905.

Add them together: 0.03675 + 0.0090 + 0.00145 + 0.000138 + 0.0000059 ≈ 0.04734.

Thus, the probability that at least 6 out of 10 recover is approximately 0.047 when rounded to three decimals, which corresponds to about 0.047.

Key ideas highlighted:
- The binomial distribution models the number of recoveries in a fixed-size cohort with independent trials.
- At least 6 is a tail probability computed by summing the relevant k values.
- Each term uses the combination factor C(10,k) and the binomial probability mass function (p^k)(1−p)^{n−k}.

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