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Kathy苏苏 · 2023年03月14日

交集和独立

  1. FRM I
  2. Quantitative

NO.PZ2020010301000005

问题如下：

Based on the probabilities in the plot below, what are the values of the following?

a. Pr(AC)Pr(A^C)

b. Pr(D|A∪B∪C)

c. Pr(A|A)

d. Pr(B|A)

e. Pr(C|A)

f. Pr(D|A)

g. Pr((A∪D)C)Pr({(A∪D)}^C)

h. Pr((AC⋂DC))Pr((A^C\bigcap D^C))

i. Are any of the four events pairwise independent?

选项：

解释：

a. 1 - Pr(A) = 100% - 30% = 70%

b. This value is Pr(D∩(A∪B∪C))/Pr(A∪B∪C). The total probability in the three areas A, B, and C is 73%. The overlap of D with these three is 9% + 8% + 7% = 24%, and so the conditional probability is 24%/73%= 33%.

c. This is trivially 100%.

d. Pr(B∩A) = 9%. The conditional probability is 9%/30% = 30%.

e. There is no overlap and so Pr(C∩A) = 0.

f. Pr(D∩A) = 9%. The conditional probability is 30%.

g. This is the total probability not in A or D. It is 1 – Pr(A∪D) = 1 - (Pr(A) + Pr(D) - Pr(A∩D)) = 100% - (30% + 36% - 9%) = 43%.

h. This area is the intersection of the space not in A with the space not in D. This area is the same as the area that is not in A or D, Pr((A∪D)C)Pr({(A\cup D)}^C) and so 43%.

i. The four regions have probabilities A = 30%, B = 30%, C = 28% and D = 36%. The only region that satisfied the requirement that the joint probability is the product of the individual probabilities is A and B because Pr(A∩B) = 9% = Pr(A)Pr(B) = 30% * 30%.

老师，AB有交集和AB是不是独立，之前是个什么关系？谢谢。

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品职答疑小助手雍 · 2023年03月14日

同学你好，AB相互独立的话，意味着两个事件不相关（但是有可能同时发生），所以会有交集。

但是AB有交集却不一定反推出AB相互独立。

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