Question

The tree diagram relates the weather forecast and school closings. Find the probability (as a percent) that the school is closed tomorrow if there is a
95
95% chance for heavy snow.

Answer 1

To accurately calculate the probability of school closures related to a weather event such as heavy snow, we need to establish a few key pieces of information which are typically found in a tree diagram:

1. The probability of heavy snow (given as 95%).
2. The probability of school closure given that heavy snow occurs (let’s denote it as \( P(S|H) \)).
3. The probability of school closure given that it does not snow (let’s denote it as \( P(S|N) \)).

Assuming the schools are expected to remain open when there's no heavy snow, we'll set \( P(S|N) \) to a low value.

Let’s consider the following example values for clarity:
- \( P(H) = 0.95 \) (Probability of heavy snow)
- \( P(N) = 0.05 \) (Probability of no heavy snow, since it must sum to 1)
- \( P(S|H) = 1 \) (Probability of school closure given heavy snow)
- \( P(S|N) = 0.1 \) (Probability of school closure given no snow)

Using the Law of Total Probability, the probability that the school is closed, \( P(S) \), is given by:

\[
P(S) = P(S|H) \cdot P(H) + P(S|N) \cdot P(N)
\]

Substituting in the values provided:

\[
P(S) = (1) \cdot (0.95) + (0.1) \cdot (0.05)
\]
\[
P(S) = 0.95 + 0.005
\]
\[
P(S) = 0.955
\]

This means the probability that the school is closed is \( 0.955 \), or 95.5%.

To express this as a percent, we convert \( 0.955 \) to a percentage:

\[
P(S) \times 100 = 95.5\%
\]

Thus, the probability that the school is closed tomorrow, assuming the values chosen for demonstration hold true, would be **95.5%**.

If you have different values or PMFs for closure based on snow conditions, please share, and we can adjust the calculations accordingly.