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Chapter 4: Q105SE (page 194)

Two machines that produce wine corks, the first one having a normal diameter distribution with mean value \({\rm{3\;cm}}\) and standard deviation\({\rm{.1\;cm}}\), and the second having a normal diameter distribution with mean value \({\rm{3}}{\rm{.04\;cm}}\) and standard deviation\({\rm{.02\;cm}}\). Acceptable corks have diameters between \({\rm{2}}{\rm{.9}}\) and\({\rm{3}}{\rm{.1\;cm}}\). If \({\rm{60\% }}\) of all corks used come from the first machine and a randomly selected cork is found to be acceptable, what is the probability that it was produced by the first machine?

Short Answer

Expert verified

The probability is \({\rm{P( machine 1}}\mid {\rm{ acceptable ) = 0}}{\rm{.5062 = 50}}{\rm{.62\% }}\).

Step by step solution

01

Definition

Probability simply refers to the likelihood of something occurring. We may talk about the probabilities of particular outcomes鈥攈ow likely they are鈥攚hen we're unclear about the result of an event. Statistics is the study of occurrences guided by probability.

02

Given in question

Given:

\({\rm{P(machine1) = 60\% = 0}}{\rm{.60}}\)

\({{\rm{X}}_{\rm{1}}}\)has a normal distribution with

\(\begin{array}{l}{{\rm{\mu }}_{\rm{1}}}{\rm{ = 3}}\\{{\rm{\sigma }}_{\rm{1}}}{\rm{ = 0}}{\rm{.1}}\end{array}\)

\({{\rm{X}}_{\rm{2}}}\)has a normal distribution with

\(\begin{array}{l}{{\rm{\mu }}_{\rm{2}}}{\rm{ = 3}}{\rm{.04}}\\{{\rm{\sigma }}_{\rm{2}}}{\rm{ = 0}}{\rm{.02}}\end{array}\)

03

Using complement rule

Complement rule:

\({\rm{P(notA) = 1 - P(A)}}\)

General multiplication rule:

\({\rm{P(A and B) = P(A) \times P(B}}\mid {\rm{A) = P(B) \times P(A}}\mid {\rm{B)}}\)

Addition rule for disjoint or mutually exclusive events:

\({\rm{P(A or B) = P(A) + P(B)}}\)

Definition Conditional probability:

\({\rm{P(B}}\mid {\rm{A) = }}\frac{{{\rm{P(A and B)}}}}{{{\rm{P(A)}}}}\)

The standardized score is the value \({\rm{x}}\) decreased by the mean and then divided by the standard deviation.

\(\begin{array}{l}{\rm{z = }}\frac{{{\rm{x - \mu }}}}{{\rm{\sigma }}}{\rm{ = }}\frac{{{\rm{2}}{\rm{.9 - 3}}}}{{{\rm{0}}{\rm{.1}}}}{\rm{\gg - 1}}{\rm{.00}}\\{\rm{z = }}\frac{{{\rm{x - \mu }}}}{{\rm{\sigma }}}{\rm{ = }}\frac{{{\rm{3}}{\rm{.1 - 3}}}}{{{\rm{0}}{\rm{.1}}}}{\rm{\gg 1}}{\rm{.00}}\\{\rm{z = }}\frac{{{\rm{x - \mu }}}}{{\rm{\sigma }}}{\rm{ = }}\frac{{{\rm{2}}{\rm{.9 - 3}}{\rm{.04}}}}{{{\rm{0}}{\rm{.02}}}}{\rm{\gg - 7}}{\rm{.00}}\\{\rm{z = }}\frac{{{\rm{x - \mu }}}}{{\rm{\sigma }}}{\rm{ = }}\frac{{{\rm{3}}{\rm{.1 - 3}}{\rm{.04}}}}{{{\rm{0}}{\rm{.02}}}}{\rm{\gg 3}}{\rm{.00}}\end{array}\)

04

Determining the probability

Determine the probability that a cork is acceptable for each machine using the normal probability table in the appendix:

\(\begin{array}{l}{\rm{P( acceptable }}\mid {\rm{ machine 1) = P}}\left( {{\rm{2}}{\rm{.9 < }}{{\rm{X}}_{\rm{1}}}{\rm{ < 3}}{\rm{.1}}} \right)\\{\rm{ = P( - 1}}{\rm{.00 < Z < 1}}{\rm{.00)}}\\{\rm{ = P(Z < 1}}{\rm{.00) - P(Z < - 1}}{\rm{.00)}}\\{\rm{ = 0}}{\rm{.8413 - 0}}{\rm{.1587}}\\{\rm{ = 0}}{\rm{.6826}}\end{array}\)

\({\rm{P(acceptable}}\mid {\rm{machine2) = P}}\left( {{\rm{2}}{\rm{.9 < }}{{\rm{X}}_{\rm{2}}}{\rm{ < 3}}{\rm{.1}}} \right){\rm{ = P( - 7}}{\rm{.00 < Z < 3}}{\rm{.00)}}\)

\({\rm{ = P(Z < 3}}{\rm{.00) - P(Z < - 7}}{\rm{.00)\gg 0}}{\rm{.9987 - 0 = 0}}{\rm{.9987}}\)

Use the complement rule:

\({\rm{P( machine 2) = 1 - P( machine 1) = 1 - 0}}{\rm{.60 = 0}}{\rm{.40}}\)

05

Calculation for probability

Use the general multiplication rule:

\(\begin{array}{c}{\rm{P( acceptable and machine 1) = P( machine 1)P( acceptable }}\mid {\rm{ machine 1)}}\\{\rm{ = 0}}{\rm{.60 \times 0}}{\rm{.6826}}\\{\rm{ = 0}}{\rm{.40956}}\end{array}\)

\({\rm{P(}}\)acceptable and machine \({\rm{2) = P(}}\)machine \({\rm{2) P(}}\) acceptable \(\mid \)machine \({\rm{2)}}\)

\({\rm{ = 0}}{\rm{.40 \times 0}}{\rm{.9987 = 0}}{\rm{.39948}}\)

Add the corresponding probabilities:

\({\rm{P(}}\)acceptable \({\rm{) = P(}}\) acceptable and machine \({\rm{1) + P(}}\) acceptable and machine \({\rm{2)}}\)

\({\rm{ = 0}}{\rm{.40956 + 0}}{\rm{.39948 = 0}}{\rm{.80904}}\)

Use the definition of conditional probability:

\(\begin{array}{c}{\rm{P( machine 1}}\mid {\rm{ acceptable ) = }}\frac{{{\rm{P( acceptable and machine 1)}}}}{{{\rm{P( acceptable )}}}}\\{\rm{ = }}\frac{{{\rm{0}}{\rm{.40956}}}}{{{\rm{0}}{\rm{.80904}}}}\\{\rm{\gg 0}}{\rm{.5062}}\\{\rm{ = 50}}{\rm{.62\% }}\end{array}\)

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Most popular questions from this chapter

In each case, determine the value of the constant\({\rm{c}}\)that makes the probability statement correct. a.\({\rm{\Phi (c) = }}{\rm{.9838}}\)b.\({\rm{P(0}} \le {\rm{Z}} \le {\rm{c) = }}{\rm{.291}}\)c.\({\rm{P(c}} \le {\rm{Z) = }}{\rm{.121}}\)d.\({\rm{P( - c}} \le {\rm{Z}} \le {\rm{c) = }}{\rm{.668}}\)e.\({\rm{P(c}} \le {\rm{|Z|) = }}{\rm{.016}}\)

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