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The problem asks to determine the probability that a house has 3 televisions (X = 3) given that it has at least 1 air conditioner (Y ≥ 1). This is a conditional probability problem.

Let's break it down step by step.

### Step 1: Identify the conditional probability formula

We want to find:

\[
P(X = 3 \mid Y \geq 1) = \frac{P(X = 3 \text{ and } Y \geq 1)}{P(Y \geq 1)}
\]

Where:
- \(P(X = 3 \text{ and } Y \geq 1)\) is the probability that the house has 3 televisions and at least 1 air conditioner.
- \(P(Y \geq 1)\) is the probability that the house has at least 1 air conditioner.

### Step 2: Compute \(P(X = 3 \text{ and } Y \geq 1)\)

From the given joint probability table, we look for the probabilities where \(X = 3\) and \(Y \geq 1\), which are the cells corresponding to \(X = 3\) and \(Y = 1, 2, 3, 4\):

- \(P(X = 3, Y = 1) = 0.15\)
- \(P(X = 3, Y = 2) = 0.09\)
- \(P(X = 3, Y = 3) = 0.07\)
- \(P(X = 3, Y = 4) = 0.05\)

Now, sum these probabilities:

\[
P(X = 3 \text{ and } Y \geq 1) = 0.15 + 0.09 + 0.07 + 0.05 = 0.36
\]

### Step 3: Compute \(P(Y \geq 1)\)

To find \(P(Y \geq 1)\), we sum all the probabilities where \(Y \geq 1\). This involves summing all the values in the table except those in the \(Y = 0\) column:

\[
P(Y \geq 1) = (0.05 + 0.09 + 0.06 + 0.01) + (0.03 + 0.05 + 0.02 + 0.06) + (0.09 + 0.07 + 0.05 + 0.02) + (0.07 + 0.01 + 0.05 + 0) = 0.67
\]

### Step 4: Apply the conditional probability formula

Now we can calculate the conditional probability:

\[
P(X = 3 \mid Y \geq 1) = \frac{0.36}{0.67} \approx 0.5373
\]

### Conclusion:
The answer closest to this value is **0.5833**, so that would be the correct choice.

---

Do you want more details on this process or have any questions?

### Related Questions:
1. How do you calculate the marginal probability for \( X = 2 \)?
2. What would the conditional probability be if we looked at \( X = 4 \) and \( Y \geq 2 \)?
3. How do you interpret joint probability tables in more complex settings?
4. What are the steps to calculate the expected value of \( X \)?
5. How would you visualize this type of joint probability data?

### Tip:
Always check that the sum of all probabilities in a joint probability table equals 1 to confirm the table is valid.

Determine the probability that a house has 3 televisions if there is at least 1 air conditioner.

Learn how to calculate the probability that a house has 3 televisions given that it has at least 1 air conditioner. This problem involves key probability concepts including joint and conditional probability, with solutions from a joint probability table.

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