Probability: Poisson distribution involving customer arrivals

[sanctifier]

Homework Statement

There are two stores A and B.

Customers can equally enter one of the two stores, i.e., for a specific customer, the probabilities she enters store A or B both are 0.5.

If the total number of customers in two stores has the Poisson distribution of parameter λ, then

Question 1: What is the probability that the number of customers in store A is non-zero and store B has no customers;

Question 2: What is the probability that the number of customers in store A is exactly 2 and store B has no customers.

Homework Equations

Poisson distribution: p(x)=λ^x/x! * e^-λ

The Attempt at a Solution

Answer 1: (1-p(0))*(0.5+0.5²+0.5³+...)=1-e^-λ
Comment: The probability that there are some customer in some store is 1 - p(0), then the probability that x customers entered store A is 0.5^x, hence their product should yield the desired answer?

Answer 2: p(2)*0.5²
Comment: Similar strategy as answer 1.

Are these correct?

Thank you in advance!

 

[Ray Vickson]

Homework Statement

There are two stores A and B.

Customers can equally enter one of the two stores, i.e., for a specific customer, the probabilities she enters store A or B both are 0.5.

If the total number of customers in two stores has the Poisson distribution of parameter λ, then

Question 1: What is the probability that the number of customers in store A is non-zero and store B has no customers;

Question 2: What is the probability that the number of customers in store A is exactly 2 and store B has no customers.

Homework Equations

Poisson distribution: p(x)=λ^x/x! * e^-λ

The Attempt at a Solution

Answer 1: (1-p(0))*(0.5+0.5²+0.5³+...)=1-e^-λ
Comment: The probability that there are some customer in some store is 1 - p(0), then the probability that x customers entered store A is 0.5^x, hence their product should yield the desired answer?

Answer 2: p(2)*0.5²
Comment: Similar strategy as answer 1.

Are these correct?

Thank you in advance!

You answer to (2) is correct, but your answer to (1) is not.

 

[sanctifier]

Ray Vickson, thank you for your replay.

If answer (2) is correct, then answer (1) is p(1)*0.5 + p(2)*0.5² + p(3)*0.5³ + ...

Is this correct?

If it is, then is there a concise form of the summation?

 

[Ray Vickson]

Ray Vickson, thank you for your replay.

If answer (2) is correct, then answer (1) is p(1)*0.5 + p(2)*0.5² + p(3)*0.5³ + ...

Is this correct?

If it is, then is there a concise form of the summation?

Yes, and yes. For the latter, see https://www.efunda.com/math/exp_log/series_exp.cfm .