A site is composed of 60% sand and 40% silt in separate layers and pockets. At this site, 10% of sand samples and 5% of silt samples are contaminated with trace amounts of arsenic. If a soil sample is selected at random, what is the probability that it is a contaminated sand sample? Also, if a soil sample was contaminated, what is the probability that the soil type is silt?
Solution
Let $A$ be the event the sample is sand.
Let $B$ be the event the sample is silt.
Let $C$ be the event the sample is contaminated
Given that $P(A)= 0.6$, $P(B) =0.4$, $P(C|A) =0.1$ and $P(C|B) = 0.05$
The probability that it is a contaminated sand sample is $P(A\cap C)$.
Thus
$$ \begin{aligned} P(A\cap C) &=P(A) P(C|A)\\ &= 0.6\times 0.1\\ &= 0.06 \end{aligned} $$
If a soil sample was contaminated, the probability that the soil type is silt is
$$ \begin{aligned} P(B|C) &= \frac{P(C|B)P(B)}{P(C|A)P(A) + P(C|B)P(B)}\\ &= \frac{0.05\times 0.4}{0.1\times 0.6 +0.05\times 0.4}\\ &=\frac{0.02}{0.08}\\ &=0.25 \end{aligned} $$
Further Reading
- Statistics
- Descriptive Statistics
- Probability Theory
- Probability Distribution
- Hypothesis Testing
- Confidence interval
- Sample size determination
- Non-parametric Tests
- Correlation Regression
- Statistics Calculators
