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:
- The probability of heavy snow (given as 95%).
- The probability of school closure given that heavy snow occurs (let’s denote it as \( P(S|H) \)).
- 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.