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Choose machine 1 with probability 1/(t+2) and machine 2 otherwise. Failure rates are 1/10 and 1/5. Find total failure probability.

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

高三選擇性必修第三册(A版).pdf · 7.1 · PDF 49 / printed page 44

TOPIC 01

Conditional and total probability

Use conditional denominators, disjoint partitions and Bayes inversion.

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

01 / Standard#Your turn

t=9

The conditioning event must have positive probability; a total-probability partition is exhaustive and disjoint.

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.

P(F)=P(M_1)P(F|M_1)+P(M_2)P(F|M_2)

Worked solution

Express the event with a conditional probability or a disjoint partition.
Apply the stated relation and retain its conditions.
$P(F)=\frac1{11}\frac1{10}+\frac{10}{11}\frac15=\frac{21}{110}$
The machine cases form a disjoint exhaustive partition.

The requested value is 0.190909090909.

Checks and common pitfalls: The machine cases form a disjoint exhaustive partition.

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

Teacher preparation and assessment

Defining relation: Use conditional denominators, disjoint partitions and Bayes inversion.
$P(A|B)=P(A\cap B)/P(B)$
Conditions: The conditioning event must have positive probability; a total-probability partition is exhaustive and disjoint.