Finding Probabilities from the Tables Revision Notes for Leaving Cert Mathematics

Finding Probabilities from the Tables

Overview

When working with the normal distribution, probabilities are often calculated using Z-scores and the Z-table. The Z-table provides cumulative probabilities for Z-scores, representing the area under the normal curve to the left of a given Z-value.

Z-Table Diagram

Steps to Find Probabilities Using the Z-Table

Step 1: Standardise the Value (Calculate Z-Score):

Use the Z-score formula:

Z = \frac{X - \mu}{\sigma}

Where:

$X$ : the value of interest.
$\mu$ : the mean.
$\sigma$ : the standard deviation.

Step 2: Locate the Z-Score in the Table:

Find the row corresponding to the first two digits of the Z-score (e.g., for 1.36, look at 1.3).
Find the column corresponding to the hundredths place (e.g., for 1.36, look under 0.06).
The intersection gives the cumulative probability $P(Z \leq z)$

Step 3: Interpret the Probability:

The cumulative probability represents the area under the curve to the left of the Z-score.

Step 4: Adjust for Other Scenarios:

To find $P(Z > z)$ : Subtract the cumulative probability from 1.

P(Z > z) = 1 - P(Z \leq z)

To find $P(a < Z < b)$ : Subtract the cumulative probability of $Z = a$ from $Z = b$

P(a < Z < b) = P(Z \leq b) - P(Z \leq a)

Worked Examples

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Example 1: Finding $P(Z \leq 1.25)$

Problem: What is the probability of a Z-score less than 1.25?

Solution:

Step 1: Locate $Z = 1.25$ in the Z-table.
Step 2: Row = 1.2, Column = 0.05.
Step 3: Intersection value = $0.8944$ .

Answer: $P(Z \leq 1.25) = 0.8944$ , or 89.44%.

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Example 2: Finding $P(Z > -0.75)$

Problem: What is the probability of a Z-score greater than -0.75?

Solution:

Step 1: Locate $Z = -0.75$ in the Z-table.

For negative Z-scores, find the cumulative probability directly.
$P(Z \leq -0.75) = 0.2266$

Step 2: Use the complement rule:

P(Z > -0.75) = 1 - P(Z \leq -0.75)

= 1 - 0.2266 = 0.7734

Answer: $P(Z > -0.75) = 0.7734$ , or 77.34%.

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Example 3: Finding $P(-1 < Z < 1.5)$

Problem: What is the probability of a Z-score between -1 and 1.5?

Solution:

Step 1: Find $P(Z \leq 1.5)$

Row = 1.5, Column = 0.00.
Value = 0.9332.

Step 2: Find $P(Z \leq -1)$

Value = $0.1587$

Step 3: Subtract:

P(-1 < Z < 1.5) = P(Z \leq 1.5) - P(Z \leq -1)

= 0.9332 - 0.1587 = 0.7745

Answer: $P(-1 < Z < 1.5) = 0.7745$ , or 77.45%.

Summary

Use the Z-score formula to standardise values:

Z = \frac{X - \mu}{\sigma}

The Z-table gives the cumulative probability to the left of a Z-score.
Adjust using rules for complements and differences to find probabilities for other scenarios.
Key rules include:
- $P(Z > z) = 1 - P(Z \leq z)$
- $P(a < Z < b) = P(Z \leq b) - P(Z \leq a)$

Finding Probabilities from the Tables (Leaving Cert Mathematics): Revision Notes