A roulette wheel has 38 slots, numbered 0,00,and 1 to 36. the slots 0 and 00 are colored green, 18 of the others are red and 18 are black. The dealer spins the wheel and at the same time rolls a small ball along the wheel in the opposite direction. The wheel is carefully balanced so that the ball is equally likely to land in any slot when the wheel slows. Gamblers can bet on various combinations of numbers and colors.
A) If you her on "red" you win if the ball lands in a red slot. What is the probability of winning with a bet on red in a single play of roulette?
B) You decide to play roulette 4 times, each time betting on red. What is the distribution of X, the number of times you win?
C) If you bet the same amount on each play and win on exactly 2 of the 4 plays, you will "break even". What is the probability that you will break even?
D) IF you win on fewer than 2 of the 4 plays, you will lose money. What is the probability that you will loose money?
what did you get?
A) The probability of winning with a bet on red in a single play of roulette can be calculated by dividing the number of red slots by the total number of slots.
There are 18 red slots out of a total of 38 slots (including 0 and 00).
So, the probability of winning with a bet on red is 18/38 ≈ 0.4737 or 47.37%.
B) To find the distribution of X, the number of times you win out of 4 plays, we can use the binomial distribution.
The probability of winning in a single play is p = 18/38 = 0.4737.
The number of trials is n = 4.
The distribution of X can be calculated using the binomial probability formula:
P(X=k) = (n choose k) * p^k * (1-p)^(n-k)
For k=0, 1, 2, 3, 4:
P(X=0) = (4 choose 0) * (0.4737)^0 * (1-0.4737)^(4-0)
P(X=1) = (4 choose 1) * (0.4737)^1 * (1-0.4737)^(4-1)
P(X=2) = (4 choose 2) * (0.4737)^2 * (1-0.4737)^(4-2)
P(X=3) = (4 choose 3) * (0.4737)^3 * (1-0.4737)^(4-3)
P(X=4) = (4 choose 4) * (0.4737)^4 * (1-0.4737)^(4-4)
Calculating each of these probabilities:
P(X=0) = (4 choose 0) * 1 * 0.5279^4 ≈ 0.1097 or 10.97%
P(X=1) = (4 choose 1) * 0.4737 * 0.5279^3 ≈ 0.3774 or 37.74%
P(X=2) = (4 choose 2) * 0.4737^2 * 0.5279^2 ≈ 0.3375 or 33.75%
P(X=3) = (4 choose 3) * 0.4737^3 * 0.5279^1 ≈ 0.1469 or 14.69%
P(X=4) = (4 choose 4) * 0.4737^4 * 0.5279^0 ≈ 0.0286 or 2.86%
So, the distribution of X is approximately:
X=0: 10.97%
X=1: 37.74%
X=2: 33.75%
X=3: 14.69%
X=4: 2.86%
C) To calculate the probability of breaking even (winning on exactly 2 out of 4 plays), we need to sum the probabilities of winning twice and multiply by the probability of losing twice.
P(winning twice) = P(X=2) ≈ 0.3375
P(losing twice) = P(X=0) ≈ 0.1097
The probability of breaking even is then P(winning twice) * P(losing twice):
P(breaking even) = P(X=2) * P(X=0) ≈ 0.3375 * 0.1097 ≈ 0.037 or 3.7%
So, the probability of breaking even is approximately 3.7%.
D) To find the probability of losing money (winning on fewer than 2 out of 4 plays), we need to sum the probabilities of winning once and winning zero times.
P(winning once) = P(X=1) ≈ 0.3774
P(winning zero times) = P(X=0) ≈ 0.1097
The probability of losing money is then P(winning once) + P(winning zero times):
P(losing money) = P(X=1) + P(X=0) ≈ 0.3774 + 0.1097 ≈ 0.4871 or 48.71%
So, the probability of losing money is approximately 48.71%.
A) To find the probability of winning with a bet on red in a single play of roulette, we need to determine the number of red slots on the wheel and divide it by the total number of slots on the wheel.
Since there are 18 red slots out of 38 total slots, the probability of winning with a bet on red is 18/38 or approximately 0.474.
B) To find the distribution of X, the number of times you win when betting on red in 4 plays, we need to consider the probability of each outcome.
The number of times you win, X, can range from 0 to 4. To calculate the probability for each value of X, we can use the binomial distribution formula.
P(X = x) = (nCx) * p^x * q^(n-x)
Where n is the number of trials (4 in this case), x is the number of successes (0, 1, 2, 3, or 4), p is the probability of success (0.474), and q is the probability of failure (1 - p).
Using this formula, we can calculate the probabilities for each value of X:
P(X = 0) = (4C0) * (0.474^0) * (0.526^4) ≈ 0.087
P(X = 1) = (4C1) * (0.474^1) * (0.526^3) ≈ 0.314
P(X = 2) = (4C2) * (0.474^2) * (0.526^2) ≈ 0.391
P(X = 3) = (4C3) * (0.474^3) * (0.526^1) ≈ 0.184
P(X = 4) = (4C4) * (0.474^4) * (0.526^0) ≈ 0.024
So the distribution of X, the number of times you win when betting on red in 4 plays, is approximately:
P(X = 0) ≈ 0.087
P(X = 1) ≈ 0.314
P(X = 2) ≈ 0.391
P(X = 3) ≈ 0.184
P(X = 4) ≈ 0.024
C) To calculate the probability of breaking even, we need to find the probability of winning exactly 2 out of the 4 plays.
Using the binomial distribution formula, we can calculate the probability:
P(X = 2) = (4C2) * (0.474^2) * (0.526^2) ≈ 0.391
So the probability of breaking even by winning exactly 2 out of the 4 plays is approximately 0.391.
D) To calculate the probability of losing money, we need to find the probability of winning fewer than 2 out of the 4 plays.
P(losing money) = P(X = 0) + P(X = 1)
Using the values from part B:
P(losing money) = 0.087 + 0.314 ≈ 0.401
So the probability of losing money is approximately 0.401.