LEARN · EXPLAIN · REVISE

X~N(t,4), Φ(1)=0.8413. Find P(t−2≤X≤t+2).

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 / Foundation#Your turn

t=8

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

Use this intermediate relation.

\Phi(-1)=1-\Phi(1)

Worked solution

Express the event with a conditional probability or a disjoint partition.
Apply the stated relation and retain its conditions.
$P(-1\le Z\le1)=2\Phi(1)-1=0.6826$
Subtract the lower tail rather than adding two cumulative probabilities.

The requested value is 0.6826.

Checks and common pitfalls: Subtract the lower tail rather than adding two cumulative probabilities.

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.