the average amount of miles on a car before it dies is 200,000 miles.
average amount of miles on a car before it dies?
200,000 miles
na
na
tsikogiannopoulos (2012) presented a different way to do these calculations. of course, it is by definition correct to assign equal probabilities to the events that the other envelope contains double or half that amount in envelope a. so the "switching argument" is correct up to step 6. given that the player's envelope contains the amount a, he differentiates the actual situation in two different games: the first game would be played with the amounts (a, 2a) and the second game with the amounts (a/2, a). only one of them is actually played but we don't know which one. these two games need to be treated differently. if the player wants to compute his/her expected return (profit or loss) in case of exchange, he/she should weigh the return derived from each game by the average amount in the two envelopes in that particular game. in the first case the profit would be a with an average amount of 3a/2, whereas in the second case the loss would be a/2 with an average amount of 3a/4. so the formula of the expected return in case of exchange, seen as a proportion of the total amount in the two envelopes, is:
two envelopes problem