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Chapter 2: Q14E (page 65)

Suppose that 55% of all adults regularly consume coffee,45% regularly consume carbonated soda, and 70% regularly consume at least one of these two products.

a. What is the probability that a randomly selected adult regularly consumes both coffee and soda?

b. What is the probability that a randomly selected adult doesn鈥檛 regularly consume at least one of these two products?

Short Answer

Expert verified

a. The probability that a randomly selected adult regularly consumes both coffee and soda is 0.30

b. The probability that a randomly selected adult doesn鈥檛 regularly consume at least one of these two products is 0.30

Step by step solution

01

Given information

The probability that all adults regularly consume coffee is 0.55.

The probability that adults regularly consume carbonated soda is 0.45.

The probability that the adults regularly consume at least one of the two products is 0.70

02

Compute the probability

Let A be the event representing that all adults regularly consume coffee.

Let B be the event representing that adults regularly consume carbonated soda.

The probabilities from the provided information are,

\(\begin{aligned}P\left( A \right) = 0.55\\P\left( B \right) = 0.45\\P\left( {A \cup B} \right) = 0.70\end{aligned}\)

a.

The probability that a randomly selected adult regularly consumes both coffee and soda is computed as,

\(\begin{aligned}P\left( {A \cap B} \right) &= P\left( A \right) + P\left( B \right) - P\left( {A \cup B} \right)\\ &= 0.55 + 0.45 - 0.70\\ &= 0.30\end{aligned}\)

Therefore, the probability that a randomly selected adult regularly consumes both coffee and soda is 0.30.

b.

The probability that a randomly selected adult doesn鈥檛 regularly consume at least one of these two products is computed as,

\(\begin{aligned}P\left( {A \cup B} \right)' &= 1 - P\left( {A \cup B} \right)\\ &= 1 - 0.70\\ &= 0.30\end{aligned}\)

Therefore, the probability that a randomly selected adult doesn鈥檛 regularly consume at least one of these two products is 0.30.

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Most popular questions from this chapter

Consider the system of components connected as in the accompanying picture. Components \({\rm{1}}\) and \({\rm{2}}\) are connected in parallel, so that subsystem works iff either 1 or 2 works; since \({\rm{3}}\)and\({\rm{4}}\) are connected in series, that subsystem works iff both\({\rm{3}}\)and\({\rm{4}}\)work. If components work independently of one another and P(component i works) \({\rm{ = }}{\rm{.9}}\)for \({\rm{i = }}{\rm{.1,2}}\)and \({\rm{ = }}{\rm{.8}}\)for \({\rm{i = 3,4}}\),calculate P(system works).

A production facility employs 10 workers on the day shift, \({\rm{8}}\) workers on the swing shift, and 6 workers on the graveyard shift. A quality control consultant is to select \({\rm{5}}\) of these workers for in-depth interviews. Suppose the selection is made in such a way that any particular group of \({\rm{5}}\) workers has the same chance of being selected as does any other group (drawing \({\rm{5}}\) slips without replacement from among \({\rm{24}}\)).

a. How many selections result in all \({\rm{5}}\) workers coming from the day shift? What is the probability that all 5selected workers will be from the day shift?

b. What is the probability that all \({\rm{5}}\) selected workers will be from the same shift?

c. What is the probability that at least two different shifts will be represented among the selected workers?

d. What is the probability that at least one of the shifts will be unrepresented in the sample of workers?

The accompanying table gives information on the type of coffee selected by someone purchasing a single cup at a particular airport kiosk.

Small

Medium

Large

Regular

\(14\% \)

\(20\% \)

\(26\% \)

Decaf

\(20\% \)

\(10\% \)

\(10\% \)

Consider randomly selecting such a coffee purchaser.

a. What is the probability that the individual purchased a small cup? A cup of decaf coffee?

b. If we learn that the selected individual purchased a small cup, what now is the probability that he/she chose decaf coffee, and how would you interpret this probability?

c. If we learn that the selected individual purchased decaf, what now is the probability that small size was selected, and how does this compare to the corresponding unconditional probability of (a)?

For any events \({\rm{A}}\) and \({\rm{B}}\) with \({\rm{P(B) > 0}}\), show that \({\rm{P(A}}\mid {\rm{B) + P}}\left( {{{\rm{A}}'}\mid {\rm{B}}} \right){\rm{ = 1}}\).

Consider randomly selecting a single individual and having that person test drive \({\rm{3}}\) different vehicles. Define events \({{\rm{A}}_{\rm{1}}}\), \({{\rm{A}}_{\rm{2}}}\), and \({{\rm{A}}_{\rm{3}}}\) by

\({{\rm{A}}_{\rm{1}}}\)=likes vehicle #\({\rm{1}}\)\({{\rm{A}}_{\rm{2}}}\)= likes vehicle #\({\rm{2}}\)\({{\rm{A}}_{\rm{3}}}\)=likes vehicle #\({\rm{3}}\)Suppose that\({\rm{ = }}{\rm{.65,}}\)\({\rm{P(}}{{\rm{A}}_3}{\rm{)}}\)\({\rm{ = }}{\rm{.70,}}\)\({\rm{P(}}{{\rm{A}}_{\rm{1}}} \cup {{\rm{A}}_{\rm{2}}}{\rm{) = }}{\rm{.80,P(}}{{\rm{A}}_{\rm{2}}} \cap {{\rm{A}}_{\rm{3}}}{\rm{) = 40,}}\)and\({\rm{P(}}{{\rm{A}}_{\rm{1}}} \cup {{\rm{A}}_{\rm{2}}} \cup {{\rm{A}}_{\rm{3}}}{\rm{) = }}{\rm{.88}}{\rm{.}}\)

a. What is the probability that the individual likes both vehicle #\({\rm{1}}\)and vehicle #\({\rm{2}}\)?

b. Determine and interpret\({\rm{p}}\)(\({{\rm{A}}_{\rm{2}}}\)|\({{\rm{A}}_{\rm{3}}}\)).

c. Are \({{\rm{A}}_{\rm{2}}}\)and \({{\rm{A}}_{\rm{3}}}\)independent events? Answer in two different ways.

d. If you learn that the individual did not like vehicle #\({\rm{1}}\), what now is the probability that he/she liked at least one of the other two vehicles?
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