A side bet offered by many casinos during a game of black is called the \"Dare a

Question

A side bet offered by many casinos during a game of black is called the "Dare any Pair Bet." In this bet if the players is dealt a pair for the first two cards, the player wins $55. Otherwise loses $5. The probability of a pair for a player's first two cards in a deck blackjack game is 0.0768. There is no need to include any amount that the players pays to play the game. a. Find the expected value for the side bet. (rounded to 2 decimal places.) b. Choose the interpretation A.) On average, the player wins $0.93 each time they make this side bet. B.) On average, the player wins $0.56 each time they make this side bet. C.) On average the player loses $0.39 each time they make this side bet. D.) On average the player loses $50.56 each time they make this side bet. C. Calculator the variance and standard deviation of the random variable X, the amount the player wins.

Explanation / Answer

Expected value can be calculated as Sum of

Payoff(probability of gain) +(Loss) (probability of loss)

55(.0768)+(-5)(1-.0768)

=4.224-4.616= -. 392

Theinterpretation would be C(looses. 39 in side bet). This is the expected value.

Now, for variance, letscalculate E(X^2) first.

=(55^2) (. 0768)+(-5)^2 (1-.0768)

232.32+ 23.08=255.4

Now, variance=E(X^ 2)-E(X)^2

255.4-(-.392)^2

255.4-.1537

=255.2463

Standard deviation is square root of variance and it would be 15.9764

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