Please answer all parts of question 1 (A-C) and show work so I can learn. Thank

Question

Please answer all parts of question 1 (A-C) and show work so I can learn. Thank you.

Conditional Probability and Independence Question 1 A) A light bulb company has factories in two cities. The factory in city A produces two-thirds of the company's light bulbs. The remainder are produced in city B, and of these, 1% are defective. Among all bulbs manufactured by the company, what proportion are not defective and made in city B? B) Suppose P(rain today) = 40% P(rain tomorrow) = 50% Prin today and tomorrow)-30% Given that it rains today, what is the chance that it will rain tomorrow? C) Two independent events have probabilities 0.1 and 0.3. What is the probability that (i) neither of the events occurs? (ii) at least one of the events occurs? ii) exactly one of the events occurs?

Explanation / Answer

1) proportion not defective and made in city B =P(B)*P(not defective|B) =(1/3)*(1-0.01) =0.33~33%

2)P(rain tomorrow|rain today) =P(rain today and tomorrow)/P(rain today) =(0.3/0.4) =0.75 ~75%

C) P( neither of events occurs) =(1-0.1)*(1-0.3)=0.63

(ii) P(at least one of the event occurs) =1-P(none of the event occurs )=1-0.63=0.37

(iii) P( exactly one event occurs) =P(first one occurs*second one does not+first one does not*second one does)

=0.1*(1-0.3)+(1-0.1)*0.3=0.34

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