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Probability refers to the likelihood of a random event's outcome. This word refers to determining the likelihood of a given occurrence occurring. What are the chances of receiving a head when we toss a coin in the air, for example? The amount of alternative outcomes determines the answer to this question. In this case, the conclusion might be either head or tail. As a result, the chance of getting a head is 1/2.

The study of data collection, analysis, interpretation, presentation, and organization is known as statistics. It is a process of gathering and analyzing data. This has a wide range of uses, from little to huge. Stats are employed in every data analysis, whether it is the study of the country's population or its economy.

Given:

\(\begin{aligned}{{\rm{P}}\left( {{{\rm{A}}_{\rm{1}}}} \right){\rm{ = 0}}{\rm{.12}}}\\{{\rm{P}}\left( {{{\rm{A}}_{\rm{2}}}} \right){\rm{ = 0}}{\rm{.07}}}\\{{\rm{P}}\left( {{{\rm{A}}_{\rm{3}}}} \right){\rm{ = 0}}{\rm{.05}}}\\{{\rm{P}}\left( {{{\rm{A}}_{\rm{1}}}{\rm{E }}{{\rm{A}}_{\rm{2}}}} \right){\rm{ = 0}}{\rm{.13}}}\\{{\rm{P}}\left( {{{\rm{A}}_{\rm{1}}}{\rm{E }}{{\rm{A}}_{\rm{3}}}} \right){\rm{ = 0}}{\rm{.14}}}\\{{\rm{P}}\left( {{{\rm{A}}_{\rm{2}}}{\rm{E }}{{\rm{A}}_{\rm{3}}}} \right){\rm{ = 0}}{\rm{.10}}}\\{{\rm{P}}\left( {{{\rm{A}}_{\rm{1}}}{\rm{{C}}}{{\rm{A}}_{\rm{2}}}{\rm{{C}}}{{\rm{A}}_{\rm{3}}}} \right){\rm{ = 0}}{\rm{.01}}}\end{aligned}\)

The complement rule is as follows.The occurrence \({{\rm{A}}_{\rm{1}}}\) refers to a kind 1 defect.

\({\rm{P}}\left( {{\rm{not}}{{\rm{A}}_{\rm{1}}}} \right){\rm{ = 1 - P}}\left( {{{\rm{A}}_{\rm{1}}}} \right){\rm{ = 1 - 0}}{\rm{.12 = 0}}{\rm{.88 = 88\% }}\)

The following is the general addition rule for any two events:

\({\rm{P(A{C}B) = P(A) + P(B) - P(A E B)}}\)

A type 1 flaw is classified as \({{\rm{A}}_{\rm{1}}}\), whereas a type 2 defect is classified as \({{\rm{A}}_{\rm{2}}}\)\({\rm{P}}\left( {{{\rm{A}}_{\rm{1}}}{\rm{{C}}}{{\rm{A}}_{\rm{2}}}} \right){\rm{ = P}}\left( {{{\rm{A}}_{\rm{1}}}} \right){\rm{ + P}}\left( {{{\rm{A}}_{\rm{2}}}} \right){\rm{ - P}}\left( {{{\rm{A}}_{\rm{1}}}{\rm{E }}{{\rm{A}}_{\rm{2}}}} \right){\rm{ = 0}}{\rm{.12 + 0}}{\rm{.07 - 0}}{\rm{.13 = 0}}{\rm{.06 = 6\% }}\)

If the system is in the intersection of \({{\rm{A}}_{\rm{1}}}\) and \({{\rm{A}}_{\rm{2}}}\)but not in the intersection of all three events \({{\rm{A}}_{\rm{i}}}\), the system has both type 1 and type 2 faults but no type 3 defect.

\({\rm{P}}\left( {{{\rm{A}}_{\rm{1}}}{\rm{{C}}}{{\rm{A}}_{\rm{2}}}{\rm{{C}A}}_{\rm{3}}^{\rm{c}}} \right){\rm{ = P}}\left( {{{\rm{A}}_{\rm{1}}}{\rm{{C}}}{{\rm{A}}_{\rm{2}}}} \right){\rm{ - P}}\left( {{{\rm{A}}_{\rm{1}}}{\rm{{C}}}{{\rm{A}}_{\rm{2}}}{\rm{{C}}}{{\rm{A}}_{\rm{3}}}} \right){\rm{ = 0}}{\rm{.06 - 0}}{\rm{.01 = 0}}{\rm{.05 = 5\% }}\)

The complement rule is as follows:

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

If the system does not fall in the intersection of all three occurrences, it has at least two flaws.\({{\rm{A}}_{\rm{i}}}\)\({\rm{P(\;At most\;2\;defects\;) = 1 - P}}\left( {{{\rm{A}}_{\rm{1}}}{\rm{{C}}}{{\rm{A}}_{\rm{2}}}{\rm{{C}}}{{\rm{A}}_{\rm{3}}}} \right){\rm{ = 1 - 0}}{\rm{.01 = 0}}{\rm{.99 = 99\% }}\)