500 adults were asked whether or not they will get married in the next 3 years. Their responses are recorded in the table.

Would marry in the next 3 years

. | Yes (Y) | No (N) |
---|---|---|

Male (M) | 150 | 150 |

Female (F) | 100 | 100 |

If one adult is selected at random from these 500 persons, find the probability that

i. the response is 'Yes';

ii. the response is 'Yes' given female;

iii. the response is 'No' and a male;

iv. the response is 'Yes' or female.

Are the events 'Male' and 'Yes' independent? Explain why or why not.

Are the events 'Female' and 'No' mutually exclusive? Explain why or why not.

### Solution

i. The probability that the response is 'Yes' is

`$P(\text{Yes}) = \frac{150+100}{500} = \frac{250}{500}= 0.5$`

.

ii. The probability that the response is 'Yes' given female is

` $$ \begin{aligned} P(\text{Yes}|\text{female}) &= \frac{P(\text{Yes}\cap \text{female})}{P(\text{female})}\\ &=\frac{100/500}{200/500}\\ &=\frac{100}{200}\\ &= 0.5 \end{aligned} $$ `

iii. The probability that the response is 'No' and male is

` $$ \begin{aligned} P(\text{No}\cap\text{male}) &= \frac{150}{500}\\ &=0.3 \end{aligned} $$ `

iv. The probability that the response is 'Yes' or female is

` $$ \begin{aligned} P(\text{Yes}\cup\text{female}) &= P(\text{Yes})+P(\text{female})-P(\text{Yes}\cap \text{female})\\ &=\frac{150+100}{500} + \frac{100+100}{500} - \frac{100}{500}\\ &= \frac{250}{500}+ \frac{200}{500}-\frac{100}{500}\\ &= \frac{350}{500}\\ &=0.7 \end{aligned} $$ `

- The probability that selected adult is male is
`$P(\text{male})= \frac{300}{500}= 0.6$`

.

The probability that the response is 'Yes' is

`$P(\text{Yes})= \frac{150+100}{500}=\frac{250}{500} = 0.5$`

.

The probability that the response is 'Yes' and male is `$P(\text{Yes}\cap \text{male}) = \frac{150}{500}= 0.3$`

.

Since `$P(\text{Yes}\cap \text{male}) = P(\text{Yes})\times P(\text{male})$`

, the events 'Male' and 'Yes' are independent.

- The probability that selected adult is female and the response is 'No' is
`$P(\text{Female}\cap \text{No})= \frac{100}{500}= 0.2\neq 0$`

. Thus the events 'Female' and 'No' are not mutually exclusive.

#### Further Reading

- Statistics
- Descriptive Statistics
- Probability Theory
- Probability Distribution
- Hypothesis Testing
- Confidence interval
- Sample size determination
- Non-parametric Tests
- Correlation Regression
- Statistics Calculators