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- Let A and B be independent events on the probability space (Ω, ?, P). Prove that the complement of A (or Ac) and the complement of B (which is Bc) are also independent.Recently, Hector ran for president of the local school board. Of those who voted in the election, 80% had a high school diploma, and the other 20% did not. Hector got 65% of the vote of those with high school diplomas, while he got only 25% of the vote of those without high school diplomas. Let D denote the event that a randomly chosen voter (in the school board election) had a high school diploma and D denote the event that a randomly chosen voter did not have a high school diploma. Let H denote the event that a randomly chosen voter voted for Hector and H denote the event that a randomly chosen voter did not vote for Hector. Fill in the probabilities to complete the tree diagram below, and then answer the question that follows. Do not round any of your responses. (If necessary, consult a list of formulas.)Prove
- Suppose that A, B and C are three independent events such that P(A)=0.2, P(B)=0.5 and P(C)=0.1. Determine the probabilty that none of these events occurs. Answer in procent.Let A be the event a person has blond hair and B be the event that a person has blue eyesLet A and B be two events with P(A and Bc) = 1. Find P(B). Recall that Bc denotes the complement of the event B.
- Let B₁..... B, be independent events. Show P(UB)=1-1-PC P(B₁)). imla) Show that for any three events A, B, and C, the probability that at least one of them occurs is P(A) + P(B) + P(C) - P(An B)- P(ANC) - P(BNC) + P(An BnC). b) Given A and B are independent events, with P(A) = 0.50 and P(B) = 0.30. Find P(ANB) c) About 52% of the residents of Capricorn municipality are happy and 48% of the residents are not happy with the delivery service. A recent study showed that 75% of happy residents and 25% of the unhappy residents are in favour of keeping the mayor of the municipality. If resident is randomly selected from the municipality residents is found to favour the motion. What is the probability that this person is happy with the delivery service?b) Events A and B are such that P(A) = 3/5, P(B) = 4/15. Find P(A U B) if i) A and B are mutually exclusive events. ii) A and B are independent events.
