A supermarket collected data to estimate the proportion of shoppers that buy certain types of products. It was found that 50% of shoppers bought dairy products, 30% bought meat, and 40% bought neither.
A shopper is selected at random. Let $A$ be the event they "bought dairy products", and let $B$ be the event they "bought meat". Let $\overline{A}$ and $\overline{B}$ denote the complements of $A$ and $B$, respectively.
i) Fill in the probabilities in a copy of the following two-way table:
| . | $B$ | $\overline{B}$ | Total |
|---|---|---|---|
| $A$ | |||
| $\overline{A}$ | |||
| Total | 1 |
ii) Find the probability of the event "bought dairy products or meat"
iii) Find the probability that a shopper buys dairy products given they do not buy meat.
iv) Are events A and B mutually exclusive? Explain clearly why or why not.
v) Are events A and B independent? Explain clearly why or why not.
Solution
i) Given that $P(A) = 0.50$, so $P(\overline{A}) = 1- P(A) = 0.50$.
$P(B) = 0.30$, so $P(\overline{B}) = 1- P(B) = 0.70$.
$P(\overline{A}\cap \overline{B}) = 0.40$.
Using all these values and calculating remaining probabilities we have
| . | $B$ | $\overline{B}$ | Total |
|---|---|---|---|
| $A$ | 0.20 | 0.30 | 0.50 |
| $\overline{A}$ | 0.10 | 0.40 | 0.50 |
| Total | 0.30 | 0.70 | 1 |
ii) The probability of the event "bought dairy products or meat" is
$$ \begin{aligned} P(A\cup B) &= P(A)+P(B) -P(A\cap B)\\ &= 0.50 +0.30 - 0.20 \\ &= 0.6. \end{aligned} $$
iii) The probability that a shopper buys dairy products given they do not buy meat is
$$ \begin{aligned} P(A|\overline{B}) &= \frac{P(A\cap \overline{B})}{P(\overline{B})}\\ &= \frac{0.30}{0.70}\\ &= 0.4286. \end{aligned} $$
iv) Events $A$ and $B$ are not mutually exclusive, because $P(A\cap B) = 0.20 \neq 0$.
v) $P(A\cap B) = 0.20$, $P(A) =0.50$, $P(B) = 0.30$.
So $P(A)\times P(B) = 0.50*0.30 = 0.15$.
Events $A$ and $B$ are not independent because $P(A\cap B) \neq P(A)\times P(B)$.
Further Reading
- Statistics
- Descriptive Statistics
- Probability Theory
- Probability Distribution
- Hypothesis Testing
- Confidence interval
- Sample size determination
- Non-parametric Tests
- Correlation Regression
- Statistics Calculators
