On a certain day, a foundry manufactured 520 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

Minor Flaw No Minor Flaw
Major Flaw 20 39
No Major Flaw 75 386

(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 both parts of the problem, we will analyze the given data in the table format. The data represents the distribution of parts based on the presence of major and minor flaws:

\[ \begin{array}{|c|c|c|} \hline & \text{Minor Flaw} & \text{No Minor Flaw} \ \hline \text{Major Flaw} & 20 & 39 \ \hline \text{No Major Flaw} & 75 & 386 \ \hline \end{array} \]

From the table, we can calculate the different categories of flaws:

1. Parts with both major and minor flaws (20):
2. Parts with major flaws but no minor flaws (39):
3. Parts with minor flaws but no major flaws (75):
4. Parts with no flaws (386):

Now, let's fill in the parts based on the table and find the total number of parts with flaws.

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

To find the total number of parts with flaws, we sum the parts that have major flaws and those that have minor flaws.

Calculating:

• Parts with major flaws = parts with both major & minor flaws + parts with major but no minor flaws = \(20 + 39 = 59\).
• Parts with minor flaws = parts with both major & minor flaws + parts with minor but no major flaws = \(20 + 75 = 95\).

However, since parts with both major and minor flaws are counted in both categories, we cannot simply add them together. Instead, we can find the total flaws directly:

Therefore, the total number of parts that have at least one type of flaw (major or minor): \[ \text{Total with flaws} = \text{Major Flaws only} + \text{Minor Flaws only} + \text{Both} = 20 + 39 + 75 = 134 \]

To find the probability of having a flaw, we calculate: \[ P(\text{Flaw}) = \frac{\text{Number of parts with flaws}}{\text{Total number of parts}} = \frac{134}{520} \]

Calculating the fraction: \[ P(\text{Flaw}) = \frac{134}{520} \approx 0.2577 \text{ (or about 25.77%)}. \]

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

To find the number of parts with no major flaw, we can refer to the 'No Major Flaw' row of the table.

From the data:

• Parts with no major flaw = parts with minor flaws but no major flaws + parts with no flaws at all = \(75 + 386 = 461\).

To find the probability of having no major flaw, we calculate: \[ P(\text{No Major Flaw}) = \frac{\text{Number of parts with no major flaws}}{\text{Total number of parts}} = \frac{461}{520} \]

Calculating the fraction: \[ P(\text{No Major Flaw}) = \frac{461}{520} \approx 0.8856 \text{ (or about 88.56%)}. \]

Summary:

(c) The probability that a randomly chosen part has a flaw (major or minor) is approximately 0.2577 (25.77%).
(d) The probability that a randomly chosen part has no major flaw is approximately 0.8856 (88.56%).

Answer 2

(a) Find the probability that a randomly chosen part has a major flaw. Round your answer to four decimal places.

(b) Find the probability that a randomly chosen part has a minor flaw. Round your answer to four decimal places.

Answer 3

To find the probabilities that a randomly chosen part has a major flaw and a minor flaw, we will use the information given in the table and the total number of parts manufactured.

From the table, we have the following distribution: \[ \begin{array}{|c|c|c|} \hline & \text{Minor Flaw} & \text{No Minor Flaw} \ \hline \text{Major Flaw} & 20 & 39 \ \hline \text{No Major Flaw} & 75 & 386 \ \hline \end{array} \]

Total number of manufactured parts:

The total number of parts is given as 520.

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

To find the probability that a part has a major flaw, we need to count all parts with a major flaw (including those with and without minor flaws):

• The total number of parts with major flaws: \[ \text{Parts with major flaws} = \text{Both major and minor flaws} + \text{Major flaws only} = 20 + 39 = 59 \]

The probability of selecting a part with a major flaw is: \[ P(\text{Major Flaw}) = \frac{\text{Number of parts with major flaws}}{\text{Total number of parts}} = \frac{59}{520} \]

Calculating this value: \[ P(\text{Major Flaw}) \approx 0.1135 \]

So, rounding to four decimal places: \[ P(\text{Major Flaw}) = 0.1135 \]

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

To find the probability that a part has a minor flaw, we need to count all parts with a minor flaw (including those with and without major flaws):

• The total number of parts with minor flaws: \[ \text{Parts with minor flaws} = \text{Both major and minor flaws} + \text{Minor flaws only} = 20 + 75 = 95 \]

The probability of selecting a part with a minor flaw is: \[ P(\text{Minor Flaw}) = \frac{\text{Number of parts with minor flaws}}{\text{Total number of parts}} = \frac{95}{520} \]

Calculating this value: \[ P(\text{Minor Flaw}) \approx 0.1827 \]

So, rounding to four decimal places: \[ P(\text{Minor Flaw}) = 0.1827 \]

Summary:

(a) The probability that a randomly chosen part has a major flaw is 0.1135.
(b) The probability that a randomly chosen part has a minor flaw is 0.1827.