Question

On a certain day, a foundry manufactured 516 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.
Major Flaw w/ minor flaw - 18
major flaw with no minor flaw- 30
no major flaw with minor flaw- 80
no major flaw with no minor flaw- 388

(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 solve this problem, we will use the data provided in the table to calculate the probabilities.

### Given data:
- Major flaw with minor flaw: 18
- Major flaw with no minor flaw: 30
- No major flaw with minor flaw: 80
- No major flaw with no minor flaw: 388

### Total parts manufactured:
Total = 516

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

To find the probability of having a major flaw, we need to find the total number of parts with a major flaw, which includes both those with and without minor flaws.

Total with major flaw = Major flaw with minor flaw + Major flaw with no minor flaw
Total with major flaw = 18 + 30 = 48

Probability of major flaw \( P(M) \):
\[
P(M) = \frac{\text{Total with major flaw}}{\text{Total parts}} = \frac{48}{516}
\]

Calculating this:
\[
P(M) \approx 0.0930
\]

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

To find the probability of having a minor flaw, we need to find the total number of parts with a minor flaw, which includes both those with and without major flaws.

Total with minor flaw = Major flaw with minor flaw + No major flaw with minor flaw
Total with minor flaw = 18 + 80 = 98

Probability of minor flaw \( P(m) \):
\[
P(m) = \frac{\text{Total with minor flaw}}{\text{Total parts}} = \frac{98}{516}
\]

Calculating this:
\[
P(m) \approx 0.1905
\]

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

To find the probability of having a flaw (either major or minor), we can find the total number of parts with at least one flaw. This includes parts with major flaws, minor flaws, or both.

Using the principle of inclusion-exclusion:
Total with at least one flaw = Total with major flaw + Total with minor flaw - Total with both major and minor flaws

Total with at least one flaw = 48 (major) + 98 (minor) - 18 (both)
Total with at least one flaw = 128

Probability of having at least one flaw \( P(F) \):
\[
P(F) = \frac{\text{Total with at least one flaw}}{\text{Total parts}} = \frac{128}{516}
\]

Calculating this:
\[
P(F) \approx 0.2470
\]

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

To find the probability of having no major flaw, we can simply use the total number of parts with no major flaws, which includes parts with and without minor flaws.

Total without major flaw = No major flaw with minor flaw + No major flaw with no minor flaw
Total without major flaw = 80 + 388 = 468

Probability of no major flaw \( P(M^c) \):
\[
P(M^c) = \frac{\text{Total without major flaw}}{\text{Total parts}} = \frac{468}{516}
\]

Calculating this:
\[
P(M^c) \approx 0.9070
\]

### Summary of probabilities:
(a) \( P(M) \approx 0.0930 \)

(b) \( P(m) \approx 0.1905 \)

(c) \( P(F) \approx 0.2470 \)

(d) \( P(M^c) \approx 0.9070 \)