Plz help with this question. Also can u explain what \"orthodontic work is indep

Question

Plz help with this question. Also can u explain what "orthodontic work is independent of the need for filling and extraction "mean?I made a ven diagram with 3 circles. but that part lost me.

2. A dental insurance policy covers three procedures: orthodontics, fillings, and extractions. During the life of policy the probability that policyholder needs a) orthodontic work is 0.5; b) orthodontic work or a filling.65; c) orthodontic work or an extraction is.75; d) a filling and an extraction is.125 The needs for orthodontic work is independent of the need for filling and extraction. Find the probability that the policyholder will need a filling or extraction during the life of policy (D

Explanation / Answer

2a) let probability of orthodontic; filling and extraction are O,F,E respectively

orthodontic work is independent of the need for filling and extraction "means that

P(OnF)=P(O)*P(F) and P(OnE)=P(O)*P(E)

here P(O) =0.5

from part (b) P(OUF) =0.65

P(O)+P(F)-P(O)*P(F) =0.65

0.5+P(F)-0.5*P(F) =0.65

P(F) =0.15/0.5=0.3

and from part C) P(OUE) =0.75

P(O)+P(E)-P(O)*P(E) =0.75

0.5+P(E)-0.5*P(E) =0.75

P(E) =0.25/0.5 =0.5

from part d) P(EnF) =0.125

therefore probability that policyholder will need a filling or extraction

=P(FUE) =P(F)+P(E)-P(FnE) =0.3+0.5-0.125=0.675

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