3.31 Hearts win. In a new card game, you start with a well-shuffled full deck and draw 3 cards without replacement. If you draw 3 hearts, you win $50. If you draw 3 black cards, you win $25. For any other draws, you win nothing.

  1. Create a probability model for the amount you win at this game, and find the expected winnings. Also compute the standard deviation of this distribution.
threehearts <- 13 / 52 * 12 / 51 * 11 / 50
threeblacks <- 26 / 52 * 25 / 51 * 24 /50
Outcome Probability Win EX
3 hearts 0.0129412 50 0.6470588
3 black cards 0.1176471 25 2.9411765
Other 0.8694118 0 0.0000000

\(\mu = \sum_{i=1}^{k} x_{i} P(X=x_{i})\) \(\approx\) $3.59

\(\sigma = \sum_{i=1}^{k} (x_{i}-\mu)^2 P(X=x_{i})\) \(\approx\) 9.64

  1. If the game costs $5 to play, what would be the expected value and standard deviation of the net profit (or loss)? (Hint: profit = winnings - cost; X - 5)
Outcome Probability Win EX
3 hearts 0.0129412 45 0.5823529
3 black cards 0.1176471 20 2.3529412
Other 0.8694118 -5 -4.3470588

\(\mu = \sum_{i=1}^{k} x_{i} P(X=x_{i})\) \(\approx\) $-1.41

\(\sigma =\sum_{i=1}^{k} (x_{i}-\mu)^2 P(X=x_{i})\) \(\approx\) 9.64

  1. If the game costs $5 to play, should you play this game? Explain.

No, you should not play this game because you are expected the lose $1.41 each round.