IB Maths AA/AI HL (hard). Probability, distributions, expected value. Practice question 4.

*there must be small letters y, m, w (not capital) in the first raw of probability distribution table.
Question from IB Maths Past Papers N09 2-13.
In each round of two different games Ying tosses three fair coins and Mario tosses two fair coins.
(a) The first game consists of one round. If Ying obtains more heads than Mario, she receives $5 from Mario. If Mario obtains more heads than Ying, he receives $10 from Ying. If they obtain the same number of heads, then Mario receives $2 from Ying. Determine Ying’s expected winnings.
(b) They now play the second game, where the winner will be the player who obtains the larger number of heads in a round. If they obtain the same number of heads, they play another round until there is a winner. Calculate the probability that Ying wins the game.