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Suppose that the probability that a given customer entering a bank is between 50 and 70 years of age is 5/9. a.) On a given day, compute the probability that the 7th customer entering the bank is also the 3rd customer who is between 50 and 70 years of age. b.) With 7 people present in the bank, compute the probability that 3 of them are between 50 and 70 years of age.

For the question, above how should I apply Poisson Distribution to figure out the answer, for this question is stated to be a Poisson Process?

Basileus
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1 Answers1

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No, you don't use a Poisson distribution, because this isn't talking about the number of customers in a given time interval. This question is assuming that each customer has probability $5/9$ of being from $50$ to $70$, independent of the others (that should be stated better - it also may not be true in real life, as customers often come together with others of similar age, and also the age distribution is likely to depend on time of day). It's also assuming that at least $7$ customers do enter the bank on that day. I would expect that to have quite a high probability, but not exactly $1$ (and certainly not $1$ if it's a Poisson process).

For (a) you want the probability that there are $2$ people $50$ to $70$ among the first $6$ customers (use Binomial for that) and then the $7$'th customer is also $50$ to $70$.

Robert Israel
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  • So can in a) Can I use $\left(\begin{array}{c}6-1\ 2-1\end{array}\right)(\frac{5}{9})^{2-1}(1-\frac{5}{9})^{6-2}$ as the solution (this is based on Pascal Distribution) – Basileus Nov 13 '18 at 21:59
  • I agree that Pascal distribution is the most appropriate approach for part (a). We are interested in the probability that there is a run of three successes (people aged 50 to 70) out of 7 individuals who have entered the bank. Those successes must be consecutive in the total of 7 trials (individuals who have entered the bank). Can @Robert Israel can provide clarification on why Binomial is preferred? – J_code Nov 18 '19 at 08:46
  • Who said anything about a run of three successes?
    "Consecutive" was never mentioned. It could be that the first, fourth and seventh customers are the successes.     – Robert Israel Nov 18 '19 at 13:46