Expected Value Homework Solutions 1. You roll two dice. What is ...

Let E 1 be the event that the first card is not an ace.Let E 2 be the event that the first card is an ace and the second card is not an ace.Let E 3 be the event that the first two cards are aces and the third card is not an ace.Let E 4 be the event that the first three cards are aces and the fourth card is not an ace.Let E 5 be the event that the first cards are aces and hence the fifth card is not an ace.p(E 1 ) = 4852 and v(E 1) = 1p(E 2 ) = 4 × 4852 × 51 and v(E 2) = 2p(E 3 ) = 4 × 3 × 4852 × 51 × 50 and v(E 3) = 3p(E 4 ) = 4 × 3 × 2 × 4852 × 51 × 50 × 49 and v(E 1) = 4p(E 5 ) = 4 × 3 × 2 × 1 × 4852 × 51 × 50 × 49 × 48 and v(E 5) = 1Substitute these values intoE.V. = p(E 1 ) × v(E 1 ) + p(E 2 ) × v(E 2 ) + p(E 3 ) × v(E 3 ) + p(E 4 ) × v(E 4 ) + p(E 5 ) × v(E 5 ) ≈ 1.0816315. What is the expected number of face cards (jack, queen, or king) in a three card hand drawn atrandom from a standard deck of cards?Solution:Let E 1 be the event that there are no face cardsLet E 2 be the event that there is exactly one face card.Let E 3 be the event that there are exactly two face cards.Let E 4 be the event that all three cards are face cards.p(E 1 ) = 40 C 352C 3and v(E 1 ) = 0p(E 2 ) = 12 × 40C 252C 3and v(E 2 ) = 1p(E 3 ) = 12 C 2 × 4052C 3and v(E 3 ) = 2p(E 4 ) = 12 C 352C 3and v(E 1 ) = 3