Question

Toothpaste Ken is traveling for his business. He has a new $0.85$-ounce tube of toothpaste that's supposed to last him the whole trip. The amount of toothpaste Ken squeezes out of the tube each time he brushes varies according to a Normal distribution with mean $0.13$ounces and standard deviation $0.02$ounces. If Ken brushes his teeth six times during the trip, what's the probability that he'll use all the toothpaste in the tube? Follow the four-step process.

Short answer

There is approximately $71.9\%$chance that he will use all of the toothpaste in the tube

Step-by-step solution

Step-1 Given Information

Given in the question that Toothpaste Ken is traveling for his business. He has a new $0.85$-ounce tube of toothpaste that's supposed to last him the whole trip. The amount of toothpaste Ken squeezes out of the tube each time he brushes varies according to a Normal distribution with mean $0.13$ounces and standard deviation $0.02$ounces. We have to find the probability that he'll use all the toothpaste in the tube.

Step-2 Explanation

We need to calculate the probability that he will use the entire bottle of toothpaste. Let's make a variable $T=6X$that represents the total amount of toothpaste he uses when brushing his teeth six times. Our objective is to locate $P(T\le 0.85)$

Mean and standard deviation of $T$,

localid="1649859947223" $\begin{array}{r}{U}_T=0.78\\ {\sigma }_T=0.12\end{array}$

We need to calculate the probability that he will use the entire bottle of toothpaste. That is $P(T\le 0.85)$

Let's standardize

localid="1649859952450" $$\begin{gathered} T=0.85 \\ z=\frac{x-{\mu }_T}{{\sigma }_T} \\ =\frac{0.85-0.78}{0.12} \\ \approx 0.58 \end{gathered}$$

Using the table of normal probability as a guide,

localid="1649859959103" $$\begin{gathered} P(T\le 0.85)=P(z\le 0.58) \\ =0.7190 \end{gathered}$$

There is approximately $71.9\%$chance that he will use all of the toothpaste in the tube