Determine the probability that a randomly chosen product is…
Determine the probability that a randomly chosen product is first-class, if it is known that $4\%$ of the products are defective and that $75\%$ of the non-defective products satisfy the conditions of first class.
Explanation of Method
A first-class product must first be non-defective, and then, among non-defective products, be first-class. So "first-class" is the product of two events, and by the multiplication rule its probability is $P(\text{good})\cdot P(\text{first class}\mid\text{good})$.
Step 1: Events
- $G$ = "the product is not defective": $P(G)=1-0.04=0.96$;
- $F$ = "the product is first-class". Every first-class product is non-defective, so $F\subseteq G$ and $F=GF$.
- Given: $P(F\mid G)=0.75$ (75% of the non-defective products are first-class).
Step 2: Multiplication rule
\[ P(F) = P(GF) = P(G)\,P(F\mid G) = 0.96\cdot0.75. \]
\[ \boxed{P(F) = 0.72.} \]
Step 3: In numbers
Out of $1000$ products: $40$ are defective, $960$ are good, and $75\%$ of these, i.e. $720$, are first-class. So $720/1000=0.72$. ✓
Step 4: A remark on the wording
The $75\%$ refers to the non-defective products, as in the standard version of this problem. If $75\%$ of all products were first-class, the answer would simply be $0.75$ and the $4\%$ would be irrelevant.
Result
| Answer | |
|---|---|
| Formula | $P(F)=P(G)P(F\mid G)=0.96\cdot0.75$ |
| Probability | $0.72$ |