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Q. 103.

Expert-verified

The Practice of Statistics for AP Examination

Found in: Page 337

Book edition 6th

Author(s) Daren Starnes, Josh Tabor

Pages 837 pages

ISBN 9781319113339

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Short Answer

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. $0.72$

The required answer is:

The correct option is $(c) 0.02$

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: 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.

Step 2: 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" $B .$

$P (A) = 0.9, P (B) = 0.8$

P (A \hat{a} < B^{'}) = P (A^{'}) P (B^{'})

(as

A

and

B

are independent)

$[1 - P (A)] [1 - P (B)] = (1 - 0.9) (1 - 0.8) = 0.1 (0.2) = 0.02$

Therefore the correct option is $(c)$

Most popular questions for Math Textbooks

Middle school values Researchers carried out a survey of fourth-, fifth-, and sixth-grade students in Michigan. Students were asked whether good grades, athletic ability, or being popular was most important to them. The two-way table summarizes the survey data.

Suppose we select one of these students at random.

Find P(athletic | $5^{t h}$ grade).
Use your answer from part (a) to help determine if the events “ $5^{t h}$ grade” and “athletic” are independent.

Butter side down Refer to the preceding exercise. Maria decides to test this

probability and drops $10$ pieces of toast from a $2.5$ -foot table. Only $4$ of them land butter

side down. Maria wants to perform a simulation to estimate the probability that $4$ or

fewer pieces of toast out of $10$ would land butter side down if the researchers’ $0.81$

probability value is correct.

a. Describe how you would use a table of random digits to perform the simulation.

b. Perform $3$ trials of the simulation using the random digits given. Copy the digits onto

your paper and mark directly on or above them so that someone can follow what you

did.

$29077$	$14863$	$61683$	$47052$	$62224$	$51025$
$95052$	$90908$	$73592$	$75186$	$87136$	$95761$
$27102$	$56027$	$55892$	$33063$	$41842$	$81868$

c. The dotplot displays the results of 50 simulated trials of dropping $10$ pieces of toast.

Is there convincing evidence that the researchers’ $0.81$ probability value is incorrect?

Explain your answer.

Keep on tossing The figure shows the results of two different sets of $5000$ coin tosses.

Explain what this graph tells you about chance behavior in the short run and the long run.

The security system in a house has two units that set off an alarm when motion is

detected. Neither one is entirely reliable, but one or both always go off when there is

motion anywhere in the house. Suppose that for motion in a certain location, the

probability that detector A goes off and detector B does not go off is $0.25$ , and the

probability that detector A does not go off is $0.35$ . What is the probability that detector

B goes off?

$a . 0.1 b . 0.35 c . 0.4 d . 0.65 e . 0.75$

An unenlightened gambler

a. A gambler knows that red and black are equally likely to occur on each spin of a

roulette wheel. He observes that $5$ consecutive reds have occurred and bets heavily on

black at the next spin. Asked why, he explains that “black is due.” Explain to the

gambler what is wrong with this reasoning.

b. After hearing you explain why red and black are still equally likely after $5$ reds on the

roulette wheel, the gambler moves to a card game. He is dealt $5$ straight red cards from

a standard deck with $26$ red cards and $26$ black cards. He remembers what you said and

assumes that the next card dealt in the same hand is equally likely to be red or black.

Explain to the gambler what is wrong with this reasoning.

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