Answer to Question #333777 in Statistics and Probability for Simmsivan

Question

On average, four students visits the Mathematics and Statistics tutoring centre during a 5-minute period.

(a) Calculate the probability that three students visit the centre during a 5 minute period. (3)

(b) During a ten-minute period, what is the probability that at least 4 students visit the centre? (5)

Answer 1

Let $X$ be the random variable, $X =$ number of students visiting the centre during t-minute period

$X\sim Po(\lambda t)$

(a)

$\lambda t=4$

$$P(X=3)=\dfrac{e^{-4}\cdot(4)^3}{3!}=0.195367$$

(b)

$\lambda t=8$

$$P(X\ge4)=1-P(X<4)$$

$$=1-\dfrac{e^{-8}\cdot(8)^0}{0!}-\dfrac{e^{-8}\cdot(8)^1}{1!}$$

$$-\dfrac{e^{-8}\cdot(8)^2}{2!}-\dfrac{e^{-8}\cdot(8)^3}{3!}$$

$$=0.957620$$