Answer to Question #198643 in Statistics and Probability for Rumana

Question

a) A business receives order at an average rate of 1 per minute.What is the probability of getting three orders in one minute?

b) An emergency service receives an average of 2.1 false alarms per day.What is the probability of getting four false alarms in a given day?

c) On average the demand for a certain product is four per week.If the stock at the beginning of each week is renewed so that there are always 6 in store , What is the probability of running out of stock in any week.

d) A call center has a capacity to deal with 25 calls per minute on average.what is the probability of getting 30 calls in any minute period?

e) The average time taken for a worker to assemble a certain product is 45 minutes.There are 10 workers employed to make these assemblies.What is the probability of assembling 10 units in an hour?

Answer 1

a) Let $X=$ the number of orders in one minute: $X\sim Po(\lambda)$

$$P(X=x)=\dfrac{e^{-\lambda}\cdot\lambda^x}{x!}$$

Given $\lambda=1$

$$P(X=3)=\dfrac{e^{-1}\cdot1^3}{3!}=\dfrac{1}{6e}\approx0.061313$$

b) Let $X=$ the number of false alarms per day: $X\sim Po(\lambda)$

Given $\lambda=2.1$

$$P(X=4)=\dfrac{e^{-2.3}\cdot2.3^4}{4!}\approx0.116902$$

c) Let $X=$ the number of products per week : $X\sim Po(\lambda)$

Given $\lambda=4$

$$P(X=6)=\dfrac{e^{-6}\cdot6^4}{4!}\approx0.104196$$

d) Let $X=$ the number of calls in one minute: $X\sim Po(\lambda)$

Given $\lambda=25$

$$P(X=30)=\dfrac{e^{-25}\cdot25^{30}}{30!}\approx0.045413$$

e) Let $X=$ the number of units in one hour: $X\sim Po(\lambda)$

Given $\lambda=\dfrac{4}{3}\cdot10=\dfrac{40}{3}$

$$P(X=10)=\dfrac{e^{-40/3}\cdot(\dfrac{40}{3})^{10}}{10!}\approx0.079256$$