Question

Among the cast aluminum parts manufactured on a certain day, 80% were flawless, 15% had only minor flaws, and 5% had major flaws. Find the probability that a randomly chosen part a. has a flaw (major or minor). b. has no major flaw.

Solution

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Given:

P(Flawless)=80%=0.80P(Minor flaw)=15%=0.15P(Major flaw)=5%=0.05\begin{align*} P(\text{Flawless})&=80\%=0.80 \\ P(\text{Minor flaw})&=15\%=0.15 \\ P(\text{Major flaw})&=5\%=0.05 \end{align*}

(a) Two events are disjoint\textbf{disjoint} or mutually exclusive\textbf{mutually exclusive}, if the events cannot occur at the same time.

We assume that no part can have both a major flaw and a minor flow, thus we use that the events are mutually exclusive.

Use the Addition rule\textbf{Addition rule} for disjoint or mutually exclusive events: P(AB)=P(A)+P(B)P(A\cup B)=P(A)+P(B)

P(Flaw)=P(Major Flaw or Minor flaw)=P(Major flaw)+P(Minor flaw)=0.15+0.05=0.20=20%\begin{align*} P(\text{Flaw})&=P(\text{Major Flaw or Minor flaw}) \\ &=P(\text{Major flaw})+P(\text{Minor flaw}) \\ &=0.15+0.05 \\ &=0.20 \\ &=20\% \end{align*}

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