Given: If any of the \({\rm{25}}\) rivets are defective, the seam must be redone. The rivets are on their own.
\(\begin{array}{c}P({\rm{ At least one of the rivits is defective }}) = 15\% \\ = 0.15\end{array}\)
Complement rule:
\({\rm{P(notA) = 1 - P(A)}}\)
Multiplication rule for independent events:
\({\rm{P(A and B) = P(A)*P(B)}}\)
Use the complement rule:
\({\rm{P}}\left( {{\rm{ No rivits are defective }}} \right){\rm{ = 1 - P}}\left( {{\rm{ At least one of the rivits is defective }}} \right)\)
\({\rm{ = 1 - 0}}{\rm{.15 = 0}}{\rm{.85}}\)
We can use the multiplication rule for independent events because the rivets are independent. The likelihood of a single rivet not being defective is represented by \({\rm{P}}\) (not defective).
\(\begin{array}{*{20}{l}}{{\rm{P}}\left( {{\rm{No rivets are defective}}} \right)}\\{{\rm{ = P}}\left( {{\rm{not defective}}} \right){\rm{*P}}\left( {{\rm{ not defective }}} \right){\rm{ *}}....{\rm{*P}}\left( {{\rm{ not defective }}} \right)}\end{array}\)
\({\rm{ = (P( not defective )}}{{\rm{)}}^{{\rm{25}}}}\)
We also know that the likelihood of this happening is \(0.85{\rm{ }}:\)
\({{\rm{(P( not defective ))}}^{{\rm{25}}}}{\rm{ = 0}}{\rm{.85}}\)
Take the 25th root of each side of the equation:
\({\rm{P( not defective ) = }}\sqrt({{\rm{25}}}){{{\rm{0}}{\rm{.85}}}} \approx 0.{\rm{9935203}}\)
Use the complement rule again:
\(\begin{array}{c}\rm P( defective ) &=& 1 - P( not defective ) \\ \rm &=& 1 - 0{\rm{.9935203}}\\ \rm &=& 0{\rm{.0064797}}\\ \rm &=& 0{\rm{.64797\% }}{\rm{.}}\end{array}\)
Thus, the probability that \({\rm{P( defective ) = 0}}{\rm{.64797\% }}{\rm{.}}\)
