Let be independent events such that for every

MATHEMATICS · Subjective · A Level · Mathematics · Probability · Probability

Problem

Let

E_1,E_2,\dots,E_n

n

independent events such that

P(E_i)=p

for every

i=1,2,\dots,n

. (i) Show that the probability that none of the events occurs is

(1-p)^n

. (ii) Hence obtain the probability that at least one of the events occurs. (iii) Evaluate

\displaystyle\lim_{n\to\infty} P(\text{at least one of }E_1,E_2,\dots,E_n \text{ occurs})

when

p=\dfrac{1}{n}

. (iv) Interpret the limiting value obtained in part (iii) in the context of rare events occurring in a large number of independent trials.