Show all necessary work for each problem; your work is part of the grading criteria. Precise usage of language and symbolsotations is part of the grading. 1. (6 points) A single card is to be drawn from a standard deck of cards. Given that A: The selected card is less thana 6 B: The selected cards is black Find the following assuming aces are high cards: e P(B) e P(B|A) e Determine whether events 4 and B are independent and justify how you came up with this answer. 2. (8 points) You and your friend decide to play a game where a single card is selected from the deck. If the card is less than 7, assuming aces are high cards, then they will pay you $7. If the cardisa 7 or higher, then you will pay them $10. Find the expected value of this game. From this information, if you were to play this game 10 times, how much could you expect to win or lose? Should you play this game? 2. (8 points) You and your friend decide to play a game where a single card is selected from the deck. If the card is less than 7, assuming aces are high cards, then they will pay you $7. If the cardisa 7 or higher, then you will pay them $10. Find the expected value of this game. From this information, if you were to play this game 10 times, how much could you expect to win or lose? Should you play this game? 3. (6 points) Jerry is a baseball pitcher who throws two kinds of pitches: a fastball and a knuckleball. The probahility of throwing a strike is different for each pitch. The probability of throwing a strike with a fastball is 0.76. The probability of throwing a strike with a knuckleball is 0.38. Jerry throws fastballs more frequently than he throws knuckleballs. On average, for every 10 pitches he throws, 7 of them are fastballs. Construct a probability tree that models this information. What is the probability that the pitcher will throw a strike on any given pitch