On a certain day, a foundry manufactured 596 cast aluminum parts. Some of these had major flaws, some had minor flaws, and some had both major and minor flaws. The following table represents the results.

Question

Major flaw with a minor flaw- 17
major flaw with no minor flaw- 33
no major flaw with a minor flaw- 71
no major flaw and no minor flaw- 475

(a) Find the probability that a randomly chosen part has a major flaw.

(b) Find the probability that a randomly chosen part has a minor flaw.

(c) Find the probability that a randomly chosen part has a flaw (major or minor).

(d) Find the probability that a randomly chosen part has no major flaw.

Answer 1

To find the required probabilities, let's interpret the information given and calculate accordingly.

We have the following counts from the table:

• Major flaw with a minor flaw: 17
• Major flaw with no minor flaw: 33
• No major flaw with a minor flaw: 71
• No major flaw and no minor flaw: 475

First, we can summarize the total number of parts with major flaws, minor flaws, and then find the total number of parts as already provided:

1. Total number of parts manufactured: 596

2. Parts with major flaws:

   • Major flaw with a minor flaw: 17
   • Major flaw with no minor flaw: 33
   • Total with major flaws = 17 + 33 = 50

3. Parts with minor flaws:

   • Major flaw with a minor flaw: 17
   • No major flaw with a minor flaw: 71
   • Total with minor flaws = 17 + 71 = 88

4. Parts with no major flaws:

   • No major flaw with a minor flaw: 71
   • No major flaw and no minor flaw: 475
   • Total with no major flaws = 71 + 475 = 546

Now we can calculate each of the requested probabilities.

(a) Probability that a randomly chosen part has a major flaw:

\[ P(\text{Major}) = \frac{\text{Number of parts with major flaws}}{\text{Total number of parts}} = \frac{50}{596} \]

Calculating the probability:

\[ P(\text{Major}) = \frac{50}{596} \approx 0.0836 \]

(b) Probability that a randomly chosen part has a minor flaw:

\[ P(\text{Minor}) = \frac{\text{Number of parts with minor flaws}}{\text{Total number of parts}} = \frac{88}{596} \]

Calculating the probability:

\[ P(\text{Minor}) = \frac{88}{596} \approx 0.1473 \]

(c) Probability that a randomly chosen part has a flaw (major or minor):

To find this, we can use the principle of inclusion-exclusion:

\[ P(\text{Flaw}) = P(\text{Major}) + P(\text{Minor}) - P(\text{Major and Minor}) \]

Where
\( P(\text{Major and Minor}) = \frac{17}{596} \) (because these are the parts that have both major and minor flaws).

Now substituting the values:

\[ P(\text{Flaw}) = \frac{50}{596} + \frac{88}{596} - \frac{17}{596} = \frac{50 + 88 - 17}{596} = \frac{121}{596} \]

Calculating this probability:

\[ P(\text{Flaw}) = \frac{121}{596} \approx 0.202 \]

(d) Probability that a randomly chosen part has no major flaw:

\[ P(\text{No Major}) = \frac{\text{Number of parts with no major flaws}}{\text{Total number of parts}} = \frac{546}{596} \]

Calculating this probability:

\[ P(\text{No Major}) = \frac{546}{596} \approx 0.916 \]

Summary of Results:

• (a) Probability of a major flaw: \( \approx 0.0836 \)
• (b) Probability of a minor flaw: \( \approx 0.1473 \)
• (c) Probability of any flaw (major or minor): \( \approx 0.202 \)
• (d) Probability of no major flaw: \( \approx 0.916 \)