Homework 4Note: I may not multiply to get the final answers in some (any) of these. You are expectedto multiply most things out in your ho mewo r ks.§2.252.) What is the probability of getting exactly four 1’s when you roll a die five times?We use the probability of g etting a 1 (16) and the probability of not getting a 1 (56) andfind that the answer isC(5, 4)164561.§2.316.) What is the probability of getting a face card given that the card is black?The quickest way to do this is to realize that there are 6 face cards among the 26 blackcards, so the answer is626=313.You can also do this using the conditional expectationP (card is a face card | card is black =P (card is a face card ∩ card is black)P (card is black)=6522652,which gives the same answer.32.) Studies of the racing history of a certain race horse show that the horse wins 12%of its races on dry tracks, but only 6% of its races on wet tracks. A certain race track is dry80% of the time and wet 20% of the the time.a.) What is the probability that, on a randomly chosen day, the track is wet and the horsewins?P (track is wet ∩ horse wins) = P (horse wins | track is wet) · P (track is wet)= .06 · .2 = .012b.) Wha t is the probability that, on a randomly chosen day, the track is wet and the horseloses?P (track is wet ∩ horse loses) = P (horse loses | track is wet) · P (track is wet)= (1 − .06) · .2 = .94 · .2 = .188135.) An unfair coin is tossed. The probabilities areP (heads) =34, .P (tails) =14.If heads comes up, you draw a ball from an unrn containing 4 green balls and 4 yellow balls.If tails comes up, you draw a ball from an urn containing 5 green balls and 10 yellow balls.What is the probability of drawing a green ball? A yellow ball?The probability of drawing a green ball isP (drawing green) = P (drawing green | coin is heads)P (coin is heads)+P (drawing green | coin is tails)P (coin is tails)=48·34+515·14=38+112=1124The easiest way to find the probability of drawing a yellow ball is by using what we justdid:P (drawing yellow) = 1 − P (drawing gr een ) =1324.You could do this the same way we did the green ball probability, too.46.) Suppose you toss three fair coins. You win 32 cents if all three coins come up thesame. You win 8 cents if exactly 1 head occurs, and you win 48 cents if exactly 2 headsappear. Would you pay 30 cents to play this game? What price should this game have?This is an expected value problem. If you play the game, your expected payoff (in cents)is32 · P (all coins the same) + 8 · P (exactly 1 head) + 48 · P (exactly 2 heads).So now all we have to do is fine the probabilities just stated. To do this, you use the formulafor finding the probability of getting n heads from 3 coin flips:C(3, n)2n.We’ll get that the expected payoff is32 ·28+ 8 ·38+ 48 ·38=2328= 29.So since you should expect to be paid 15 cents when playing this g ame, you should no tpay 30 cents to play it. The price of the game is therefore 15