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in a large casino, the house wins on its blackjack tables with a probability of 51.6%. all bets at blackjack are 1 to 1 - if you win, you gain the amount you bet; if you lose, you lose the amount you bet. a. what is the expected value to you of a single game? what is the house edge? b. if you played 500 games of blackjack in an evening, betting $10 on each hand, how much should you expect to win or lose? c. if patrons bet $4,000,000 on blackjack in one evening, how much should the casino expect to earn?

          in a large casino, the house wins on its blackjack tables with a probability of 51.6%. all bets at blackjack are 1 to 1 - if you win, you gain the amount you bet; if you lose, you lose the amount you bet. 
a. what is the expected value to you of a single game? what is the house edge?
b. if you played 500 games of blackjack in an evening, betting $10 on each hand, how much should you expect to win or lose?
c. if patrons bet $4,000,000 on blackjack in one evening, how much should the casino expect to earn?

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Added by Andrew W.

Elementary Statistics a Step by Step Approach

Allan G. Bluman 9th Edition

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Solved by Expert Jon Southam

04/13/2022

Step 1

- The probability that the house wins is 51.6%, which means the probability that you win is 48.4% (since the total probability must equal 100%). Show more…

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in a large casino, the house wins on its blackjack tables with a probability of 51.6%. all bets at blackjack are 1 to 1 - if you win, you gain the amount you bet; if you lose, you lose the amount you bet. a. what is the expected value to you of a single game? what is the house edge? b. if you played 500 games of blackjack in an evening, betting $10 on each hand, how much should you expect to win or lose? c. if patrons bet $4,000,000 on blackjack in one evening, how much should the casino expect to earn?

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Transcript

00:01 All right, we have a nice little expected value problem with a large casino where blackjack wins at this probability, 51 .6%.

00:11 And we're going with the assumption that everything's one -to -one, where if you win, you get the amount back, you gain the amount you bet.

00:20 But if you lose, you lose the amount you bet.

00:22 So it's pretty straightforward there.

00:26 So if you bet two bucks, let's do a nice bet.

00:29 Let's say you do five bucks, then if you win, you walk away with 10, however, you're getting plus five.

00:43 Whereas if you lose, you're getting rid of that five bucks.

00:49 So if you win, you get plus the amount you bet.

00:52 If you lose, minus the amount you bet.

00:54 So for a, what is the expected value to you, of a single game? so let's just say it's a buck.

01:02 For simplicity's sake.

01:04 Well, it's a buck times 0 .516, and we're going to add to it a buck times one minus 0 .516.

01:18 And that's going to give us 0 .032.

01:22 So what does this mean? well, let's think.

01:25 Where are we? the, where are you? most likely you're not the casino, unless you are the casino.

01:31 In which case, good for you.

01:35 But you're betting.

01:35 We're going to assume that we're the player.

01:37 So this is the net gain for the casino.

01:42 This is the casino's edge.

01:49 Whoops, subtract.

01:52 This is the casino's edge, which means they gained this amount for us.

02:01 Our loss is the negative of this.

02:07 So negative .032.

02:14 I guess this is a cent because this is a buck...

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