Question

Find the probability that her cats will wake her up no more than five times next week.

$$\begin{gathered} a.0.5000 \\ b.0.9329 \\ c.0.0378 \\ d.0.0671 \end{gathered}$$

Short answer

The answer is Option (d) is the correct answer$P(X\le 5)=0.0571$.

Step-by-step solution

Introduction 

Probability is a discipline of mathematics concerned with numerical explanations of the likelihood of an event occurring or the truth of a claim. The probability of an event is a number between 0 and 1, with 0 indicating impossibility and 1 indicating certainty, broadly speaking.

Given

Mrs. Plum's cats wake her up on average 10 times every week.

Formula used 

This is the formula we used $P(X=x)=\frac{e^{-\lambda }\times {\lambda }^x}{x!}$.

Explanation 

$P(X\le 5)=P(X=0)+P(X=1)+P(X=2)+P(X=3)+P(X=4)$

$+P(X=5)$

$P(X\le 5)=\frac{e^{-10}\times {10}^0}{0!}+\frac{e^{-10}\times {10}^1}{1!}+\frac{e^{-10}\times {10}^2}{2!}+\frac{e^{-10}\times {10}^3}{3!}+\frac{e^{-10}\times {10}^4}{4!}+\frac{e^{-10}\times {10}^5}{5!}$

it is near about given $P(X\le 5)=0.057085\approx 0.0571$

so, the final answer we have ,

We can easily see that Option D is right based on the calculations above.

$P(X\le 5)=0.0571$.