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To find the marginal probability that \( Y = 20 \), we need to sum the joint probabilities of \( Y = 20 \) for each possible value of \( Z \). The table gives us:

- \( P(Y = 20, Z = 28) = 0.4 \)
- \( P(Y = 20, Z = 32) = 0.3 \)

The marginal probability \( P(Y = 20) \) is calculated by summing these joint probabilities:

\[
P(Y = 20) = P(Y = 20, Z = 28) + P(Y = 20, Z = 32) = 0.4 + 0.3 = 0.7
\]

So, the marginal probability that \( Y = 20 \) is **0.7**.

Would you like more details, or do you have any questions about this? Here are some related questions to explore:

1. How do we compute the marginal probability \( P(Y = 10) \)?
2. What is the marginal probability distribution of \( Z \)?
3. How do we verify that the total probability sums to 1?
4. Can we determine the conditional probability \( P(Y = 20 \mid Z = 28) \)?
5. How is joint probability related to marginal and conditional probabilities?

**Tip:** Always check that the sum of all joint probabilities equals 1 to ensure the table represents a valid probability distribution.

What is the marginal probability that Y = 20, based on the joint probability table?

Learn how to calculate the marginal probability P(Y = 20) using joint probability values from a distribution table. This tutorial covers key probability concepts, including joint and marginal probabilities, and applies the law of total probability. Ideal for students in Grades 10-12.

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