A laboratory tested 200 synthetic compound samples for two properties: thermal stability a… — GMAT

A laboratory tested $200$ synthetic compound samples for two properties: thermal stability and chemical resistance. Among the samples, $120$ exhibited thermal stability, $90$ exhibited chemical resistance, and $50$ exhibited neither property. If a sample is selected at random from those that exhibited thermal stability, what is the probability that it also exhibited chemical resistance?

Answer: 0.5

Answer

0.5

The conditional probability of selecting a sample with chemical resistance given that it has thermal stability is found by dividing the number of samples with both properties (

60

) by the total number of samples with thermal stability (

120

), giving

\frac{60}{120} = 0.5

Step-by-Step Solution

Find the total number of samples exhibiting at least one of the two properties.

Since

50

out of

200

samples exhibited neither property, the number of samples exhibiting at least one property is

200 - 50 = 150

The total population consists of samples exhibiting at least one property plus samples exhibiting neither property.

Calculate the number of samples exhibiting both thermal stability (

T

) and chemical resistance (

C

Using the inclusion-exclusion principle

|T \cup C| = |T| + |C| - |T \cap C|

, we have

150 = 120 + 90 - |T \cap C|

, which yields

|T \cap C| = 60

Overlapping sets require subtracting the intersection to avoid double-counting elements.

Compute the conditional probability

P(C|T)

P(C|T) = \frac{|T \cap C|}{|T|} = \frac{60}{120} = 0.5

The given condition restricts the sample space to only the

120

samples exhibiting thermal stability.

Key Concept

Conditional probability restricts the sample space to the given condition's outcome space:

P(A|B) = \frac{|A \cap B|}{|B|}

Estimated Time:1m 30s

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