Answer:
(a) E(x) = -0.081 S.D = 3
(b) E(x) = -0.081 S.D = 1.73
(c) it is less risky to bet $1 in three different rounds as compared to betting $3 in a single round.
Step-by-step explanation:
(a) You bet $3 on a single round which means that if you win the game, your amount will double ($6), your profit will be $3. Whereas, if you lose the round, your profit will be -$3. You can only bet on red or black and both have 18 slots each.
So, the probability of landing the ball in a red/black slot = 18/37. This is the probability of winning. The probability of losing can be calculated as 1-18/37 = 19/37.
We can make a probability distribution table:
x 3 -3
P(X=x) 18/37 19/37
Expected value E(x) can be calculated as:
E(x) = ∑ x.P(x)
= (3)(18/37) + (-3)(19/37)
E(x) = -0.081
Standard deviation can be calculated by the following formula:
Var(x) = E(x²) - E(x)²
S.D = √Var(x)
We need to first calculate E(x²).
E(x²) = ∑x².P(x)
= (3)²(18/37) + (-3)²(19/37)
= (9)(18/37) + (9)(19/37)
E(x²) = 9
Var(x) = E(x²) - E(x)²
= 9 - (-0.081)²
Var(x) = 8.993
S.D = √8.993
S.D = 2.99 ≅ 3
(b) Now, the betting price is $1 and 3 rounds are played. We will compute the expectation for one round and then add it thrice to find the expectation for three rounds. Similarly, for the standard deviations, we will add the individual variances and then consider the square root of it.
E(x) = ∑ x.P(x)
= (1)(18/37) + (-1)(19/37)
E(x) = -0.027
Standard deviation can be calculated by the following formula:
Var(x) = E(x²) - E(x)²
S.D = √Var(x)
We need to first calculate E(x²).
E(x²) = ∑x².P(x)
= (1)²(18/37) + (-1)²(19/37)
= (1)(18/37) + (1)(19/37)
E(x²) = 1
Var(x) = E(x²) - E(x)²
= 1 - (-0.027)²
Var(x) = 0.9992
The expectation for one round is -0.027
For three rounds,
E(x₁ + x₂ + x₃) = E(x₁) + E(x₂) + E(x₃)
= (-0.027) + (-0.027) + (-0.027)
E(x₁ + x₂ + x₃) = -0.081
Similarly, the variance for one round is 0.9992.
Var (x₁ + x₂ + x₃) = Var(x₁) + Var(x₂) + Var(x₃)
= 0.9992 + 0.9992 + 0.9992
Var (x₁ + x₂ + x₃) = 2.9976
S.D = √2.9976
S.D = 1.73
(c) The expected values for both part (a) and (b) are the same but the standard deviation is lower in part (c) as compared to (b). Since the standard deviation is less in part (c), it means that it is <u>less risky to bet $1 in three different rounds as compared to betting $3 in a single round.</u>
