In each of the following situations, state whether or not the given assignment of probabilities to individual outcomes is legitimate, that is, satisfies the rules of probability. If not, give specific reasons for your answer.
(a) Choose a college student at random and record gender and enrollment status: P(female full-time) = 0.45, P(female part-time) = 0.55, P(male full-time) = 0.41, P(male part-time) = 0.59.
legitimatenot legitimate because the probabilities sum to 2not legitimate because P(female part-time | male part-time) does not equal 1not legitimate because P(female full-time) and P(male full-time) do not sum to 1not legitimate because P(female part-time) and P(male part-time) do not sum to 1
(b) Deal a card from a shuffled deck: P(clubs) = 12/52, P(diamonds) = 12/52, P(hearts) = 12/52, P(spades) = 16/52.
legitimate (for a nonstandard deck)not legitimate because the probabilities do not sum to 1not legitimate because P(clubs) does not equal 1/2not legitimate because P(hearts) does not equal 1/2not legitimate because the probabilities are independent
(c) Roll a die and record the count of spots on the up-face: P(1) = 1/3, P(2) = 1/6, P(3) = 0, P(4) = 1/3, P(5) = 1/6, P(6) = 0.
legitimate (for a nonstandard die)not legitimate because some probabilities are 0not legitimate because the the complement of P(4) is 1not legitimate because the probabilities are dependentnot legitimate because each probability should be 1/2