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w1,w2,w3,w4,w5 are normally distributed i.i.d R.V.

I want to calculate the probability that slots 1&2(w1,w2) are smaller than slots 3,4,5 (where mu>0)

I'm able to calculate the probability that slot 1 is smaller than 3,4,5, now i'm stuck on the calculation for slot 2, where i know that i have to consider the outcomes from the first calculation, for example if i know that w1(slot 1) is larger than w2(slot 2), than the probability that w2(slot 2) is smaller than (3,4,5) is 1.

until now i was sure that the whole probability is :

$$

P_{correct}= P(w_1<\mu+w_3)P(w_1<\mu+w_4)P(w_1<\mu+w_5)P(w_2<\mu+w_3)P(w_2<\mu+w_4)P(w_2<\mu+w_5)

$$

now i am thinking of:

$$

P_{correct}= P(w_1<\mu+w_3)P(w_1<\mu+w_4)P(w_1<\mu+w_5)P(w_2<\mu+w_3|w_1<w_2)P(w_2<\mu+w_4|w_1<w_2)P(w_2<\mu+w_5|w_1<w_2)P(w_2<\mu+w_3|w_2<w_1)P(w_2<\mu+w_4|w_2<w_1)P(w_2<\mu+w_5|w_2<w_1)

$$where the last 3 terms are equal to 1 because i have the knowledge that $$P(w1<μ+w3)P(w1<μ+w4)P(w1<μ+w5) $$

any help will be appreciated.

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# Conditional probability for normally R.V

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