LEARN · EXPLAIN · REVISE

Models A~N(t,4), B~N(t+10,9). Compare P(A≤t+2) and P(B≤t+13).

Read the idea, work independently, then explain what changed.

高三選擇性必修第三册(A版).pdf · 7.5 · PDF 88 / printed page 83

TOPIC 01

Normal distribution

Standardise a normal variable and interpret supplied table probabilities and symmetry.

Standardise a normal variable and interpret supplied table probabilities and symmetry.
Justify the method and check the conditions in a new situation.

Use one hint at a time. A correction explains what changed, not just the final answer.

01 / Transfer#Worked example

t=4

σ>0; probability table values are supplied in each task. A continuous variable has zero probability at a single point.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Express the event with a conditional probability or a disjoint partition.

Hint 2

Standardise each bound.

Worked solution

Express the event with a conditional probability or a disjoint partition.
Apply the stated relation and retain its conditions.
$z_A=2/2=1,\quad z_B=3/3=1$
Equal z-scores give equal probabilities despite different units and scales.

The requested relation or conclusion is shown below.

P(A\le\mu_A+\sigma_A)=P(B\le\mu_B+\sigma_B)

Checks and common pitfalls: Equal z-scores give equal probabilities despite different units and scales.

Reasoning checklist · self / teacher assessment

Think first. Reveal a hint when the class is ready.

Teacher preparation and assessment

Standardise a normal variable and interpret supplied table probabilities and symmetry.
Which condition is essential in normal distribution?
Does shifting the mean change the probability within one standard deviation?

Defining relation: Standardise a normal variable and interpret supplied table probabilities and symmetry.
$Z=(X-\mu)/\sigma$
Conditions: σ>0; probability table values are supplied in each task. A continuous variable has zero probability at a single point.