Question

An athlete suspected of using steroids is given two tests that operate independently of each other. Test A has probability $0.9$of being positive if steroids have been used. Test B has probability $0.8$of being positive if steroids have been used. What is the probability that neither test is positive if the athlete has used steroids?

a. $0.08$

b. $0.28$

c. $0.02$

d. $0.38$

e. $0.72$

Short answer

The required answer is:

The correct option is$\left(c\right)0.02$

Step-by-step solution

Given information

The Test $A$has the probability $0.9$of being positive if steroids have been used. The Test $B$has the probability $0.8$of being positive if steroids have been used.

Calculation

Let $A$denote the event that an athlete is given a test $A.$
Let $B$denote the event that an athlete is given a test role="math" localid="1654796054098" $B.$

$P\left(A\right)=0.9,P\left(B\right)=0.8$

$P\left(A\hat{a}<B^{\text{'}}\right)=P\left(A^{\text{'}}\right)P\left(B^{\text{'}}\right)$(as $A$ and $B$ are independent)

$$\begin{gathered} \left[1-P\left(A\right)\right]\left[1-P\left(B\right)\right] \\ =(1-0.9)(1-0.8) \\ =0.1(0.2) \\ =0.02 \end{gathered}$$

Therefore the correct option is$\left(c\right)$