It is found that 30% of the population of an island are overweight. Among the overweight, the probability of those who do not have any chronic illness is 0.4 and among those who are not overweight, the probability that they do not have any chronic illness is 0.65.

If two persons are randomly chosen from the population, find the probability that

(i) both of them do not have any chronic illness.

(ii) only one of them has chronic illness.

(iii) at least one of them does not have any chronic illness

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itzamirv2 itzamirv2 · Answer 1 · 2024-03-09T23:42:39+01:00

(i) Probability that both of them do not have any chronic illness:

[tex]P(\neg C \cap \neg C) = P(\neg C) \times P(\neg C) [/tex]

[tex]= (P(\neg C | O) \times P(O)) \times (P(\neg C | \neg O) \times P(\neg O)) [/tex]

[tex]

= (0.4 \times 0.30) \times (0.65 \times 0.70) [/tex]

[tex]= 0.12 \times 0.455 = 0.0546 [/tex]

(ii) Probability that only one of them has a chronic illness:

[tex]P((\neg C \cap C) \cup (C \cap \neg C)) = P(\neg C \cap C) + P(C \cap \neg C) [/tex]

[tex]= (P(\neg C | O) \times P(C | \neg O) \times P(O)) + (P(C | O) \times P(\neg C | \neg O) \times P(\neg O)) [/tex]

[tex]= (0.4 \times 0.35 \times 0.30) + (0.6 \times 0.65 \times 0.70)[/tex]

[tex]= 0.042 + 0.273 = 0.315 [/tex]

(iii) Probability that at least one of them does not have any chronic illness:

[tex]P(\neg C \cup \neg C) = 1 - P(C \cap C) [/tex]

[tex]= 1 - (P(C) \times P(C)) [/tex]

[tex]= 1 - ((P(C | O) \times P(O)) \times (P(C | \neg O) \times P(\neg O))) [/tex]

[tex]= 1 - ((0.6 \times 0.30) \times (0.35 \times 0.70)) [/tex]

[tex]

= 1 - (0.018 + 0.0735) [/tex]

[tex]= 1 - 0.0915 = 0.9085[/tex]

The correct probabilities are:

(i) 0.0546

(ii) 0.315

(iii) 0.9085