If a person visits his dentist, suppose the probability that he will have his teeth cleaned is 0.48, the probability that he will have a cavity filled is 0.25, the probability that he will have a tooth extracted is 0.20, the probability that he will have a teeth cleaned and a cavity filled is 0.09, the probability that he will have his teeth cleaned and a tooth extracted is 0.12, the probability that he will have a cavity filled and a tooth extracted is 0.07, and the probability that he will have his teeth cleaned, a cavity filled, and a tooth extracted is 0.03. What is the probability that a person visiting his dentist will have at least one of these things done to him?
Let A = event of teeth cleaned, B = event of cavity filled, C = event of tooth extracted.
We need P(A ∪ B ∪ C).
Using inclusion–exclusion principle:
P(A ∪ B ∪ C) = P(A) + P(B) + P(C) − [P(A ∩ B) + P(A ∩ C) + P(B ∩ C)] + P(A ∩ B ∩ C)
Substitute values:
= 0.48 + 0.25 + 0.20 − (0.09 + 0.12 + 0.07) + 0.03
= 0.93 − 0.28 + 0.03
= 0.68
Answer: The probability that a person visiting his dentist will have at least one of these things done is 0.68.