Answer:
4.6
Step-by-step explanation:
Let x represent the amount you can win
Therefore
X = (0, 5, 10, 20)
Given that
No. Of red card in a deck = 26
No. Of spade in a deck = 13
No. Of clubs in a deck less ace = 12
No. Of ace in of club = 1
Total number of cards in a deck = 52
Therefore
P(X=0) = P(probability of picking a red card) = 26/52 = 0.50
P(X=5) = P(probability of picking a spade) = 13/52 = 0.25
P(X=10) = P(proability of picking a club, but not the ace) = 12/52 ≈ 0.231
P(X=20) = P(probability of picking the ace of clubs) = 1/52 ≈ 0.019
Next, we find the expected winnings
Recall that,
Expected winnings = mean = μ = E[X] = ∑X × P(X=x)
Therefore,
μ = (0 × 26/52) + (5 × 13/52) + (10 × 12/52) + (20 × 1/52)
μ = 0 + 65/52 + 120/52 + 20/52
= 205/52
= 3.94
Finally, we find the standard deviation
Recall that,
Standard Deviation = Squareroot of the variance = σ
Variance (σ2)= V[X] = ∑X(x−μ)^2 × P(X = x)
σ2=(0−205/52)^2 × (26/52)+(5−205/52)^2 × (13/52) + (10−205/52)^2 × (12/52)+(20−205/52)^2 × (1/52)
σ2 = 42,025/5,408 + 3,025/10,816 + 297,675/35,152 + 697,225/140,608
σ2 = 87,075/10,816 + 145,225/10,816 = 232,300/10,816 = 58,075/2,704 ≈ 21.477
Thus, variance = 21.477
Therefore,
Standard Deviation = squareroot of variance
= Sqroot 21.477
= 4.6
