In the game of roulette, a steel ball is rolled onto a wheel that contains 18 red, 18 black, and 2 green slots. If the ball is rolled 24 times, find the probability of the following events.

A. The ball falls into the green slots 4 or more times.

Probability =

B. The ball does not fall into any green slots.

Probability =

C. The ball falls into black slots 11 or more times.

Probability =

D. The ball falls into red slots 12 or fewer times.

Probability =

To find the probability of each event, we first need to determine the total number of possible outcomes. In this case, since there are 18 red, 18 black, and 2 green slots, there is a total of 38 slots on the wheel.

A. The ball falls into the green slots 4 or more times.
To find the probability of this event, we need to determine the number of ways the ball can fall into the green slots 4 or more times during the 24 rolls. We can use the concept of combinations to calculate this.

The number of ways the ball can fall into the green slots 4 or more times can be calculated by adding the combinations for each possible number of green slots, which is 4, 5, 6, ..., 24.

The formula to calculate combinations is:
nCr = n! / (r!(n-r)!)
where n is the total number of trials (24 rolls) and r is the number of successful outcomes (number of times the ball falls into the green slots).

Using this formula, we can calculate the probabilities for each case and sum them up to get the total probability of the ball falling into the green slots 4 or more times.

B. The ball does not fall into any green slots.
To find the probability of this event, we need to determine the number of ways the ball can avoid falling into any green slots during the 24 rolls. Since there are 2 green slots, the number of ways the ball can land on any other color is 38 - 2 = 36.

The probability of the ball not falling into any green slots can be calculated as (36/38) * (36/38) * ... * (36/38) for 24 rolls.

C. The ball falls into black slots 11 or more times.
To find the probability of this event, we need to determine the number of ways the ball can fall into the black slots 11 or more times during the 24 rolls. Similarly to the previous cases, we can calculate this using combinations.

D. The ball falls into red slots 12 or fewer times.
To find the probability of this event, we need to determine the number of ways the ball can fall into the red slots 12 or fewer times during the 24 rolls. Again, we can calculate this using combinations.

Using the calculated combinations, we can divide them by the total number of possible outcomes (38^24) to obtain the respective probabilities for each event.

I don't know if I use the poisson function in excel. I just don't know how to start the problem.

I can find the probability of hitting a green slot in one time but not multiple times.

We will be glad to critique your thinking.

Poisson statistics is not the only way to do these problems, but it can provide an approximate result for some.

For B, the probability is that of no-green 24 times in a row. (36/38)^24 = (18/19)^24 = 0.273 That was easy

For A, add the probabilities of getting green 4,5,6...24 etc times in 24 attempts. The sum will rapidly converge.
Probability of 4 green:
(1/19)^4*(18/19)^20*C(24,4)= 0.02765
Probability of 5:
(1/19)^5*(18/19)^19*C(24,5) = 0.00614
Probability of 6:
(1/19)^6*(18/19)^18*C(24,6) = 0.00108
Probability of 7:
(1/19)^7*(18/19)^17*C(24,7) = 0.00015
Probability of 4 or more: 0.0350

If a Poisson distribution is used, for n = 24 spins with p = 1/19 probability of green each time, a = np = 1.26316
P(4) = a^4*e^-1.236/4! = 0.03082
You still have to add up P(5), P(6) etc.