Question

An athlete suspected of using steroids is given two tests that operate independently of each other. Test A has a probability $0.9$ of being positive if steroids have been used. Test B has a probability $0.8$ of being positive

if steroids have been used. What is the probability that neither test is positive if steroids have been used?

$\left(a\right)0.72\left(c\right)0.02\left(e\right)0.08\left(b\right)0.38\left(d\right)0.28$

Short answer

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

Step-by-step solution

Step 1. Given Information

$$\begin{gathered} P(TestApositive)=0.9 \\ P(TestBpositive)=0.8 \end{gathered}$$

Step 2. Concept used

Complement rule: $P(A^c)=P(notA)=1-P\left(A\right)$

Step 3. Calculation

The complement rule is $P(A^c)=P(notA)=1-P\left(A\right)$

Use the complement rule to help you:

$$\begin{gathered} P(TestAnegative)=1-P(TestApositive)=1-0.9=0.1 \\ P(TestBnegative)=1-P(TestBpositive)=1-0.8=0.2 \end{gathered}$$

According to the question, the two tests are independent of one another, hence the multiplication rule for independent events can be used:

Fill in the blanks with the corresponding probabilities:

$$\begin{gathered} P(Neithertestpositive)=P(Bothtestnegative) \\ =P(TestAnegative)\times P(TestBnegative) \\ =0.1\times 0.2 \\ =0.02 \end{gathered}$$

As a result, the proper option is $\left(c\right)0.02$ according to the option.