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.) when a coin is spun, P(H)=0.55 and P(T)=0.45.
b.) When two coins are tossed, P(HH)=0.4, P(HT)=0.4, P(TH)=0.4, and P(TT)=0.4.
a. Why would the probability of a coin landing on one side be different that landing on the other, if it is a fair coin?
b. The probability of all events occurring always = 1.
a.) This assignment of probabilities to individual outcomes is not legitimate. The sum of the probabilities of all possible outcomes must equal 1. In this case, P(H) + P(T) = 0.55 + 0.45 = 1, so the sum is valid. However, the probabilities assigned to P(H) and P(T) individually are not valid. Since flipping a fair coin should result in heads (H) and tails (T) with equal likelihood, the legitimate assignment would be P(H) = 0.5 and P(T) = 0.5.
b.) This assignment of probabilities to individual outcomes is not legitimate. Once again, the sum of the probabilities of all possible outcomes must equal 1. In this case, P(HH) + P(HT) + P(TH) + P(TT) = 0.4 + 0.4 + 0.4 + 0.4 = 1.6, which is greater than 1. Therefore, this assignment violates the rule of probability. The probabilities assigned to the outcomes HH, HT, TH, and TT individually are also not valid. Since each coin flip should result in heads (H) and tails (T) with equal likelihood, the legitimate assignment would be P(HH) = 0.25, P(HT) = 0.25, P(TH) = 0.25, and P(TT) = 0.25.