A study of checkout lines at the Safeway revealed that between 4-7pm on weekdays there is an average of 4 customers waiting in line. What is the probability that you visit that Safeway during this time period and...

1-no customers are waiting in line?
2-Four customers are waiting?
3-Four or fewer customers are waiting?
4-Four or more customers are waiting?

To determine the probabilities, we need to use the concept of the Poisson distribution. Given that the average number of customers waiting in line is 4, we can use the Poisson formula to calculate the probabilities.

1. Probability that no customers are waiting in line:
The Poisson probability is calculated using the formula P(x; λ) = (e^(-λ) * λ^x) / x!, where λ is the average number of events (customers waiting) and x is the desired number of events (0 customers waiting).

Using λ = 4 and x = 0:
P(0; 4) = (e^(-4) * 4^0) / 0! = e^(-4) ≈ 0.0183
So, the probability that no customers are waiting is approximately 0.0183 or 1.83%.

2. Probability that four customers are waiting in line:
Using λ = 4 and x = 4:
P(4; 4) = (e^(-4) * 4^4) / 4! = e^(-4) * 4^4 / 4! ≈ 0.1954
The probability of exactly four customers waiting in line is approximately 0.1954 or 19.54%.

3. Probability that four or fewer customers are waiting in line:
The probability can be calculated by summing the probabilities of having 0, 1, 2, 3, or 4 customers waiting.
P(≤4; 4) = P(0; 4) + P(1; 4) + P(2; 4) + P(3; 4) + P(4; 4)
= e^(-4) + e^(-4)*4 + e^(-4)*(4^2)/2 + e^(-4)*(4^3)/(2*3) + e^(-4)*(4^4)/(2*3*4) ≈ 0.6767
The probability that four or fewer customers are waiting in line is approximately 0.6767 or 67.67%.

4. Probability that four or more customers are waiting in line:
The probability can be calculated by subtracting the probability of having three or fewer customers from 1 (since the sum of all probabilities must equal 1).
P(≥4; 4) = 1 - P(≤3; 4)
= 1 - (P(0; 4) + P(1; 4) + P(2; 4) + P(3; 4))
= 1 - (e^(-4) + e^(-4)*4 + e^(-4)*(4^2)/2 + e^(-4)*(4^3)/(2*3))
≈ 0.3233
The probability that four or more customers are waiting in line is approximately 0.3233 or 32.33%.

So, the probabilities are:
1. Probability that no customers are waiting in line ≈ 0.0183 or 1.83%.
2. Probability that four customers are waiting in line ≈ 0.1954 or 19.54%.
3. Probability that four or fewer customers are waiting in line ≈ 0.6767 or 67.67%.
4. Probability that four or more customers are waiting in line ≈ 0.3233 or 32.33%.

To determine the probabilities, we need to use the concept of probability distribution. In this case, we have been given that there is an average of 4 customers waiting in line between 4-7pm on weekdays.

To find the probabilities for each scenario, we will need to use the Poisson distribution, as it is commonly used to model the number of events occurring in a fixed interval of time when the events are rare and independent of each other.

For the Poisson distribution, the parameter λ represents the average number of events occurring in the given time interval. In our case, λ=4.

Let's calculate the probabilities for each scenario:

1. Probability that no customers are waiting in line:
To find this probability, we use the Poisson formula:

P(X=0) = (e^(-λ) * λ^0) / 0!
= e^(-4) * 1 / 1
= e^(-4) ≈ 0.0183

So, the probability that no customers are waiting in line is approximately 0.0183.

2. Probability that four customers are waiting in line:
Similarly, we can use the Poisson formula for this scenario:

P(X=4) = (e^(-λ) * λ^4) / 4!
= e^(-4) * 4^4 / 4!
= (e^(-4) * 256) / 24
= e^(-4) * 10.6667
≈ 0.1953

So, the probability that exactly four customers are waiting in line is approximately 0.1953.

3. Probability that four or fewer customers are waiting in line:
To find this probability, we need to calculate the cumulative probabilities for X=0, X=1, X=2, X=3, and X=4, and sum them up:

P(X≤4) = P(X=0) + P(X=1) + P(X=2) + P(X=3) + P(X=4)

Using the Poisson formula, we can calculate each individual probability and sum them up. The result will be approximately 0.6288.

So, the probability that four or fewer customers are waiting in line is approximately 0.6288.

4. Probability that four or more customers are waiting in line:
To find this probability, we need to calculate the complementary probability of the previous scenario (i.e., the probability that more than four customers are waiting):

P(X≥4) = 1 - P(X≤4)

We can subtract the previous result from 1 to find the probability. The result will be approximately 0.3712.

So, the probability that four or more customers are waiting in line is approximately 0.3712.

Please note that these probabilities are based on the assumption that the arrival of customers follows a Poisson distribution during the specified time period.