91Ó°ÊÓ

A lot contains 15 castings from a local supplier and 25 castings from a supplier in the next state. Two castings are selected randomly, without replacement, from the lot of 40 Let \(A\) be the event that the first casting selected is from the local supplier, and let \(B\) denote the event that the second casting is selected from the local supplier. Determine: (a) \(P(A)\) (b) \(P(B \mid A)\) (c) \(P(A \cap B)\) (d) \(P(A \cup B)\) Suppose three castings are selected at random, without replacement, from the lot of \(40 .\) In addition to the definitions of events \(A\) and \(B,\) let \(C\) denote the event that the third casting selected is from the local supplier. Determine: (e) \(P(A \cap B \cap C)\) (f) \(P\left(A \cap B \cap C^{\prime}\right)\)

Step by step solution

01

Understanding Event (a)

Event \(A\) is the event that the first casting selected is from the local supplier. There are 15 local supplier castings and a total of 40 castings. Thus, \(P(A) = \frac{15}{40} = \frac{3}{8}\).

02

Probability of Event (b) Conditional on (a)

Given that the first casting is from the local supplier, there are now 14 local supplier castings left out of 39 total castings for the next draw. Therefore, \(P(B \mid A) = \frac{14}{39}\).

03

Probability of Intersection for Two Castings (c)

To find \(P(A \cap B)\), we multiply the probability of \(A\) by \(P(B \mid A)\): \[P(A \cap B) = P(A) \cdot P(B \mid A) = \frac{3}{8} \cdot \frac{14}{39} = \frac{42}{312} = \frac{7}{52}\].

04

Using Probability General Addition Rule (d)

To find \(P(A \cup B)\), we use the formula: \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\). First, find \(P(B) = \frac{15}{40} = \frac{3}{8}\). Then, calculate \(P(A \cup B) = \frac{3}{8} + \frac{3}{8} - \frac{7}{52} = \frac{13}{20}\).

05

Probability of Event Intersection for Three Castings (e)

For \(P(A \cap B \cap C)\), we first find \(P(A \cap B)\) as calculated in step (c). Now, the probability that the third casting is also local is \(\frac{13}{38}\). Thus, \[P(A \cap B \cap C) = P(A \cap B) \cdot P(C \mid A \cap B) = \frac{7}{52} \cdot \frac{13}{38} = \frac{91}{1976}\].

06

Probability of Two from Local, Third Not (f)

For \(P(A \cap B \cap C')\), first calculate \(P(A \cap B)\) as in step (c). Then, the probability that the third casting is not local, i.e., from the out-of-state supplier, is \(\frac{25}{38}\). Thus, \[P(A \cap B \cap C') = \frac{7}{52} \cdot \frac{25}{38} = \frac{175}{1976}\].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Conditional Probability

Conditional probability is the likelihood of an event occurring, given that another event has already occurred. It helps in understanding how probabilities change when we have some additional information. In our exercise, we had to find the conditional probability of event \(B\), given that event \(A\) has occurred, denoted as \(P(B \mid A)\).
\(B\) is the event where the second casting selected is from the local supplier, knowing the first was too.
After \(A\) happens, only 14 local castings remain from a total of 39. Thus, the probability of \(B\) after \(A\) is:

• \(P(B \mid A) = \frac{14}{39}\)

Conditional probability captures this change in our scenario, showing how information about the first casting impacts the likelihood of the second.

Probability Events

Probability events are essentially the possible outcomes within a probability experiment that we might be interested in studying. In this case, we had multiple events such as \(A\), \(B\), and \(C\), which refer to whether the selected castings are from the local supplier.
Event \(A\) occurs when the first casting is local, and its probability is:

• \(P(A) = \frac{15}{40} = \frac{3}{8}\)

Considering the overall 40 castings, 15 being local, this is straightforward. Events are the building blocks of probability; understanding and defining these clearly is crucial for accurate probability calculations.

Addition Rule

The addition rule in probability allows us to find the probability of the union of two or more events. It is useful when we want the probability of either one event happening or another.
For events \(A\) and \(B\), the formula is:

• \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\)

In our example, this formula helped us calculate the probability that at least one of the two castings is from the local supplier:

• First, \(P(B) = \frac{3}{8}\) in the same way as \(P(A)\)
• \(P(A \cap B)\) was computed earlier as \(\frac{7}{52}\)

Using these probabilities, the addition rule gives us:

• \(P(A \cup B) = \frac{3}{8} + \frac{3}{8} - \frac{7}{52} = \frac{13}{20}\)

This helps in understanding how to manage overlapping probabilities.

Intersection of Events

The intersection of events involves calculating the joint probability that multiple events occur simultaneously. This is denoted by \( P(A \cap B) \), representing the probability of both event \(A\) and event \(B\) occurring.
In our exercise, we computed \( P(A \cap B) \) for the first two castings being local. We did this by multiplying the probability of the first being local by the conditional probability that the second one is also local:

• \(P(A \cap B) = P(A) \cdot P(B \mid A) = \frac{3}{8} \cdot \frac{14}{39} = \frac{7}{52}\)

The concept of the intersection is key when dealing with simultaneous occurrences of events, making complex problems more manageable.