Let be independent events with probabilities respectively.

MATHEMATICS · Multi-part · CBSE · Mathematics · Probability · Probability

Problem

Let E1,E2,…,EnE_1, E_2, \ldots, E_n be nn independent events with probabilities p1,p2,…,pnp_1, p_2, \ldots, p_n respectively. (i) Show that the probability that none of the events E1,E2,…,EnE_1, E_2, \ldots, E_n occurs is ∏i=1n(1−pi)\prod_{i=1}^{n}(1-p_i). (ii) Hence prove that the probability that at least one of the events occurs is 1−∏i=1n(1−pi)1-\prod_{i=1}^{n}(1-p_i). (iii) Using your result in part (ii), determine the probability that at least one event occurs when n=4n=4 and p1=0.20,  p2=0.25,  p3=0.30,  p4=0.35p_1=0.20,\; p_2=0.25,\; p_3=0.30,\; p_4=0.35.

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