Question

Detecting anthrax. Researchers at the University of SouthFlorida Center for Biological Defense have developed asafe method for rapidly detecting anthrax spores in powdersand on surfaces (USF Magazine, Summer 2002). Themethod has been found to work well even when there arevery few anthrax spores in a powder specimen. Considera powder specimen that has exactly 10 anthrax spores.Suppose that the number of anthrax spores in the sampledetected by this method follows an approximate uniformdistribution between 0 and 10.

a. Find the probability that 8 or fewer anthrax spores are detected in the powder specimen.

b. Find the probability that between 2 and 5 anthrax spores are detected in the powder specimen.

Short answer

a. The probability that 8 or fewer anthrax spores are detected in the powder specimen is 0.8 .

b. The probability that between 2 and 5 anthrax spores are detected in the powder specimen is. 0.3 .

Step-by-step solution

Given information

The powder specimen has exactly 10anthrax spores.

The number of anthrax spores follows an approximate uniform distribution between 0 and 10.

Step 2:Define the probability density function

Let, X denote the number of anthrax spores

X follows a uniform distribution.

The p.d.f of X is given by,

$$\begin{gathered} f\left(x\right)=\frac{1}{d-c};c\le x\le d \\ =\frac{1}{10-0};\le x\le 10 \\ =\frac{1}{10} \end{gathered}$$

Compute the probability

The probability that 8 or fewer anthrax spores are detected in the powder specimen is $P\left(X\le 8\right)$.

$$\begin{gathered} P\left(X\le 8\right)={\int }_0^{8}f\left(x\right)dx \\ ={\int }_0^8\frac{1}{10}dx \\ =\frac{8}{10} \\ =0.8 \end{gathered}$$

Hence, the probability that 8 or fewer anthrax spores are detected in the powder specimen is 0.8 .

Compute the probability

The probability that between 2 and 5 anthrax spores are detected in the powder specimen is$P\left(2\le X\le 5\right)$ .

$$\begin{gathered} P\left(2\le X\le 5\right)={\int }_2^{5}f\left(x\right)dx \\ ={\int }_2^5\frac{1}{10}dx \\ =\frac{\left(5-2\right)}{10} \\ =\frac{3}{10} \\ =0.3 \end{gathered}$$

Hence, the probability that between 2 and 5 anthrax spores are detected in the powder specimen is. 0.3 .