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