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X~N(t,4), Φ(1)=0.8413. Find P(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=7

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.

(t+2-t)/2=1

Worked solution

Express the event with a conditional probability or a disjoint partition.
Apply the stated relation and retain its conditions.
$P(X\le\mu+\sigma)=\Phi(1)=0.8413$
The requested bound is one standard deviation above the mean.

The requested value is 0.8413.

Checks and common pitfalls: The requested bound is one standard deviation above the mean.

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.