Ms. Bentley pays 7 to play a certain game at the casino....

Ms. Bentley pays $7 to play a certain game at the casino. Each player in this game has 18% probability of winning $16. (a) What is her expected value for playing this game? Round to the nearest cent (two decimal places) and be sure to include the the sign to indicate if it is gain or loss. $ (b) If Ms. Bentley plays this game once, will she come out financially ahead, behind, or is it impossible to know ahead behind impossible to know (c) Ms. Bentley goes to the casino very often. If she plays this game many times times over the next year, will she come out financially ahead, behind, or is it impossible to know? ahead behind impossible to know

Transcript

00:01 Okay, so we know that if the game is fair, the probability a jack should come up, a jack should come up in 0 .173 of the games.

00:12 The probability that a jack would come up in a game is 0 .173.

00:15 If the game is rigged, the investigator suspects it will come up at 0 .105, lots of the games.

00:29 So part a says on the first day she observed 17 games, under the assumption the game is rigged, calculate the probability that she observes three jacks.

00:38 So we'll call x the number of jacks observed.

00:47 We'll call xr if rigged and xf the number of jacks observed if fair.

00:56 Then we know that the rigged one follows a binomial distribution for the number of games seen with probability 0 .105.

01:07 And the fair one follows the same for the number of games seen, but with probability 0 .173 of success.

01:15 So part a asks for the probability that xr, the game is rigged and she sees three jacks.

01:24 That's gonna be given by which she sees 17 games.

01:27 So n is 17.

01:31 So it's gonna be by the binomial formula, it's gonna be 17 choose three times pr to the three times one minus pr to the 17 minus three, which is 14.

01:42 And if you sub in pr equals 0 .105, you can find that this is equal to 0 .1666 to four decimal places.

02:01 Part b says if the game is fair, calculate the probability that a jack is observed in at most two of the games.

02:08 So that's the probability that xf is less than or equal to 2 again, n equals 17.

02:15 And this equals the sum from i equals zero to two of the probabilities that xf equals i, which is just the sum from i equals zero to two of 17 choose i times pf to the i times one minus pf to the 17 minus i.

02:38 And if you plug that in for, add up the probabilities that xf equals zero, one and two, you find that you get 0 .4160 to four decimal places.

02:53 Part c says if the game is rigged, what's the probability she observes at least three jacks? so the probability that if the game is rigged, she observes at least three is one minus the probability that if the game is rigged, she observes less than or equal to two.

03:08 And that's just one minus the sum from i equals zero to two of 17 choose i pr to the i one minus pr to the 17 minus i...

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