The table shows the number of male and female students enrolled in nursing at the University of Oklahoma Health Sciences Center for a recent semester. A student is selected at random. Find the probability of each event. (Adapted from University of Oklahoma Health Sciences Center Office of Institutional Research)

. | Nursing majors | Non-nursing majors | Total |
---|---|---|---|

Males | 151 | 1104 | 1255 |

Females | 1016 | 1693 | 2709 |

Total | 1167 | 2797 | 3964 |

(a) The student is male or a nursing major.

(b) The student is female or not a nursing major.

(c) The student is not female or is a nursing major.

(d) Are the events "being male" and "being a nursing major" mutually exclusive? Explain.

#### Solution

(a) The probability that the student is male or a nursing major

` $$ \begin{aligned} P(\text{Male } \cup \text{ Nursing Major}) &= P(\text{Male}) + P(\text{ Nursing Major}) \\ & \quad - P(\text{Male } \cap \text{ Nursing Major})\\ &= \frac{1255}{3964} +\frac{1167}{3964} - \frac{151}{3964}\\ &= 0.5348 \end{aligned} $$ `

(b) The probability that the student is female or not a nursing major

` $$ \begin{aligned} P(\text{Female } \cup \text{ Non-Nursing Major}) &= P(\text{Female}) + P(\text{ Non-Nursing Major})\\ &\quad - P(\text{Female } \cap \text{ Non-Nursing Major})\\ &= \frac{2709}{3964} +\frac{2797}{3964} - \frac{1693}{3964}\\ &= 0.9619 \end{aligned} $$ `

(c) The probability that the student is not female or is a nursing major

` $$ \begin{aligned} P(\text{Not Female } \cup \text{ Nursing Major}) &= P(\text{Not Female}) + P(\text{ Nursing Major})\\ &\quad - P(\text{Not Female } \cap \text{ Nursing Major})\\ &= \frac{1255}{3964} +\frac{1167}{3964} - \frac{151}{3964}\\ &= 0.5348 \end{aligned} $$ `

(d) The events "being male" and "being a nursing major" not mutually exclusive because

`$P(\text{Male} \cap \text{Nursing Major}) = \frac{151}{3964} = 0.0381 \neq 0$`

.

#### Further Reading

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