A true-false question is to be posed to a husband and wife team. Both the husband and the wife will give the correct answer with probability 0.8. Which of the following is a better strategy for the couple? Strategy 1: choose one of them at random and ask the chosen person to answer the question Strategy 2: have them both answer the question. If their answers agree, give the common answer; if they disagree, flip a fair coin to determine which answer to given.

- First note that P ( W ) & P ( H ) are independent from one another, i.e the probability of giving correct answer of husband does not influences that of wife's.

- This strategy poses an event such that either wife knows the answer and answer it correctly or the husband knows and answers in correctly.

- We will assume that probability of either the husband or wife knowing the answer is 0.5 and the two events of knowing and answering correctly are independent. So,

P ( Wk ) = P (Hk) = 0.5

- The event P(S1) is:

P(S1) = P ( Hk & H ) + P ( Wk & W )

P(S1) = 0.5*0.8 + 0.5*0.8

P(S1) = 0.8

- Hence, the probability of success for strategy 1 is = 0.8

Strategy 2

- Both agree , then the common answer is selected otherwise, one of their answers is chosen at random.

- The success of strategy 2, will occur when both agree and are correct, wife is correct and answers while husband is not or husband is correct and he answers.

- The event P(S2) is:

P(S2) = P ( H & W ) + P ( H / W' & Hk ) + P ( H' / W & Wk )

P(S2) = P ( H & W ) + P ( H / W') P ( Hk ) + P ( H' / W) P (Wk)

P(S2) = P ( H & W ) + P ( H / W')*0.5 + P ( H' / W)*0.5

P(S2) = 0.5* [ P ( H & W ) + P ( H / W') ] + 0.5* [ P ( H' / W) + P ( H & W )]

P(S2) = 0.5*P(H) + 0.5*P(W)

P(S2) = 0.5*0.8 + 0.5*0.8

P(S2) = 0.8

- Hence, the probability of success for strategy 2 is = 0.8

Both strategy give us the same probability of success.