Q. 110 - Precalculus Enhanced with Graphing Utilities

Question

People enter a line for the Demon Roller Coaster at the rate of $4$ per minute. The following formula from probability can be used to determine the probability that x people will arrive within the next minute.

$P\left(x\right)=\frac{4^x{e}^{-4}}{x!}$

where, $x!=x\times \left(x-1\right)\times \left(x-2\right)....\times 3\times 2\times 1$

Part (a): Determine the probability that $x = 5$ people will arrive within the next minute.

Part (b): Determine the probability that $x = 8$ people will arrive within the next minute.

Step-by-Step Solution

Answer

Part (a): The probability that $x=5$ people will arrive within the next minute is $0.156$.

Part (b): The probability that $x=8$ people will arrive within the next minute is$0.029$.

Part (a) Step 1. Determine the probability for 5 people within the next minute.

Assume x to be the no. of people coming.

From the given question,

$P\left(x\right)=\frac{4^x{e}^{-4}}{x!}$

Substitute $x=5$ in the given function,

$$\begin{gathered} P\left(5\right)=\frac{4^5{e}^{-4}}{5!} \\ =0.156 \end{gathered}$$

Part (b) Step 1. Determine the probability for 8 people within the next minute.

Assume x to be the no. of people coming.

From the given question,

$P\left(x\right)=\frac{4^x{e}^{-4}}{x!}$

Substitute $x=8$ in the given function,

$$\begin{gathered} P\left(8\right)=\frac{4^8{e}^{-4}}{8!} \\ =0.029 \end{gathered}$$