Solved: 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

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)

(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