Container 1 has 8 items, 3 of which are defective. Container 2 has 5 items, 2 of which are defective. If one item is drawn independently from each container:

Find the probability distribution for X defined as the number of defective items drawn(Hint: You have to use both multiplicative and additive rules to find P(X=1), where as P(X=0) and P(X=2) can be found only by multiplicative rules).

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To find the probability distribution for X, we need to calculate the probabilities of each possible outcome: drawing 0, 1, or 2 defective items.

Let's start with P(X=0), which means drawing 0 defective items.
To calculate this probability, we need to find the probability of drawing a non-defective item from both containers.

In Container 1, there are 8 items in total, with 3 defective ones. So, the probability of drawing a non-defective item from Container 1 is 5/8.

Similarly, in Container 2, there are 5 items in total, with 2 defective ones. So, the probability of drawing a non-defective item from Container 2 is 3/5.

Since the events are independent, we can use the multiplicative rule to find the joint probability of drawing a non-defective item from both containers:
P(X=0) = (5/8) * (3/5) = 15/40 = 3/8

Now, let's move on to P(X=1), which means drawing 1 defective item.

To calculate this probability, we can use both the multiplicative and additive rules:

Case 1: Drawing a defective item from Container 1 and a non-defective item from Container 2.

The probability of drawing a defective item from Container 1 is 3/8.
The probability of drawing a non-defective item from Container 2 is 3/5.
Using the multiplicative rule, the probability of this case is: (3/8) * (3/5) = 9/40.

Case 2: Drawing a non-defective item from Container 1 and a defective item from Container 2.

The probability of drawing a non-defective item from Container 1 is 5/8.
The probability of drawing a defective item from Container 2 is 2/5.
Using the multiplicative rule, the probability of this case is: (5/8) * (2/5) = 10/40 = 1/4.

Now, we can use the additive rule to find P(X=1):
P(X=1) = probability of Case 1 + probability of Case 2 = 9/40 + 1/4 = 9/40 + 10/40 = 19/40.

Finally, let's find P(X=2), which means drawing 2 defective items.
To calculate this probability, we need to find the probability of drawing a defective item from both containers:

The probability of drawing a defective item from Container 1 is 3/8.
The probability of drawing a defective item from Container 2 is 2/5.
Using the multiplicative rule, the probability of this case is: (3/8) * (2/5) = 6/40 = 3/20.

Therefore, we have P(X=2) = 3/20.

To find the probability distribution for X, we need to determine the probabilities for each possible outcome: X = 0, X = 1, and X = 2.

Let's calculate the probability for each outcome:

1. X = 0 (no defective items drawn):
- Probability of selecting a non-defective item from Container 1: 5/8 (since there are 8 items with 3 defective)
- Probability of selecting a non-defective item from Container 2: 3/5 (since there are 5 items with 2 defective)
- By the multiplicative rule, the probability of both events happening is (5/8) * (3/5) = 15/40 = 3/8
- Therefore, P(X = 0) = 3/8

2. X = 1 (one defective item drawn):
- There are two cases to consider:
- Drawing a defective item from Container 1 and a non-defective item from Container 2: (3/8) * (3/5) = 9/40
- Drawing a non-defective item from Container 1 and a defective item from Container 2: (5/8) * (2/5) = 10/40
- By the additive rule, we sum up the probabilities of these two cases: (9/40) + (10/40) = 19/40
- Therefore, P(X = 1) = 19/40

3. X = 2 (two defective items drawn):
- Probability of selecting a defective item from Container 1: 3/8
- Probability of selecting a defective item from Container 2: 2/5
- By the multiplicative rule, the probability of both events happening is (3/8) * (2/5) = 6/40
- Therefore, P(X = 2) = 6/40

Now we have the probability distribution for X:

P(X = 0) = 3/8
P(X = 1) = 19/40
P(X = 2) = 6/40