Statistical Sciences 2141A/B Lecture Notes - Lecture 8: Conditional Probability

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Two events a and b are independent if the occurrence of one has no effect on the occurrence of the other event. If you roll a fair dice 3 times and get a 6 each time, the probability of rolling it again is still. This is an example of two independent events as the events of the dice being rolled have no effect on the occurrence of the other event. Independent events can occur at the same time, but the fact that one event has occured does not affect the probability of the other event occurring. Give(cid:374) (cid:374) (cid:373)utually i(cid:374)depe(cid:374)de(cid:374)t eve(cid:374)ts, a(cid:1005) , a(cid:1006) , , a(cid:374) the pro(cid:271)a(cid:271)ility of the o(cid:272)(cid:272)urre(cid:374)(cid:272)e of all of the(cid:373) (cid:894)a(cid:1005) a(cid:374)d a(cid:1006) a(cid:374)d a(cid:374)d a(cid:374)(cid:895) is the produ(cid:272)t of the pro(cid:271)a(cid:271)ilities of the i(cid:374)dividual eve(cid:374)ts. If the probability of a first event b is p(b) and if after event b has occurred, the probability of a second event a is p(a|b), then.

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