Probability & Statistics Examples

Work through these examples to understand probability and statistics symbols.

Symbol Usage Examples

Here are common usage examples for each symbol:

Probability of A ( $P(A)$ ): The chance that event A will occur ( $0 \leq P(A) \leq 1$ )
Mu ( $\mu$ ): Population Mean (average) ( $\mu = \frac{1}{n}\sum_{i=1}^{n} x_i$ )
Sigma (lowercase) ( $\sigma$ ): Standard Deviation ( $\sigma = \sqrt{\frac{1}{n}\sum_{i=1}^{n}(x_i - \mu)^2}$ )
x-bar ( $\bar{x}$ ): Sample Mean ( $\bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_i$ )
Chi-Squared ( $\chi^2$ ): Distribution used in hypothesis testing ( $\chi^2$ test)

Probability: If you flip a fair coin, $P(\text{heads}) = 0.5$ or $P(\text{heads}) = \frac{1}{2}$ .
Mean: For data set $\{2, 4, 6, 8\}$ , the mean is $\bar{x} = \frac{2 + 4 + 6 + 8}{4} = 5$ .
Standard Deviation: Measures how spread out the data is. A smaller $\sigma$ means data points are closer to the mean.

Problem 1: If you flip a fair coin twice, what is the probability of getting two heads?

Solution:

Probability of heads on first flip: $P(H_1) = \frac{1}{2}$
Probability of heads on second flip: $P(H_2) = \frac{1}{2}$
Since flips are independent: $P(H_1 \text{ and } H_2) = P(H_1) \times P(H_2) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$

Answer: $P(\text{two heads}) = \frac{1}{4} = 0.25$

Problem 2: Calculate the mean and standard deviation for the data set: $\{2, 4, 6, 8, 10\}$

Solution:

Mean ( $\bar{x}$ ): $\bar{x} = \frac{2 + 4 + 6 + 8 + 10}{5} = \frac{30}{5} = 6$
Standard Deviation ( $\sigma$ ):
- Variance: $\sigma^2 = \frac{1}{n}\sum_{i=1}^{n}(x_i - \mu)^2$
- $(2-6)^2 = 16$ , $(4-6)^2 = 4$ , $(6-6)^2 = 0$ , $(8-6)^2 = 4$ , $(10-6)^2 = 16$
- $\sigma^2 = \frac{16 + 4 + 0 + 4 + 16}{5} = \frac{40}{5} = 8$
- $\sigma = \sqrt{8} = 2\sqrt{2} \approx 2.83$

Answer: Mean $\bar{x} = 6$ , Standard deviation $\sigma \approx 2.83$

Problem 3: In a class of 30 students, 12 are girls. What is the probability that a randomly selected student is a girl?

Solution:

Answer: $P(\text{girl}) = \frac{2}{5} = 0.4$ or $40\%$

Probability and statistics help you make informed decisions, understand data, and assess risks in everyday situations.

Probability ( $P(A)$ ): Assess weather forecasts. If the forecast says $P(\text{rain}) = 0.7$ or $70\%$ , you know there's a high chance of rain and should bring an umbrella.
Risk Evaluation: Make insurance decisions. Understanding probability helps you assess whether insurance is worth the cost based on likelihood of events.

Averages ( $\bar{x}$ , $\mu$ ): Compare product prices. Calculate the mean price across stores: $\bar{x} = \frac{\text{price}_1 + \text{price}_2 + \text{price}_3}{3}$ to find the best deal.
Standard Deviation ( $\sigma$ ): Understand price variation. A low $\sigma$ means prices are consistent; a high $\sigma$ means you should shop around more.

Probability: Understand medical test results. If a test has $95\%$ accuracy ( $P(\text{correct}) = 0.95$ ), you know there's a $5\%$ chance of error.
Averages: Track health metrics. Calculate your average heart rate ( $\bar{x}$ ) over a week to monitor fitness progress.

Probability: Make strategic decisions. In card games, calculate $P(\text{winning})$ based on cards you've seen to decide whether to bet or fold.
Statistics: Analyze performance. Calculate batting averages ( $\bar{x}$ ) or shooting percentages to evaluate player performance.

Risk Assessment: Use probability to evaluate investments. If an investment has $P(\text{profit}) = 0.6$ , you know there's a $60\%$ chance of making money.
Averages: Track portfolio performance. Calculate mean returns ( $\mu$ ) over time to assess investment strategy.

Probability: Assess product reliability. If a product has $P(\text{defect}) = 0.02$ or $2\%$ , you know $98\%$ of products work correctly.
Standard Deviation: Understand consistency. Low $\sigma$ in manufacturing means consistent quality; high $\sigma$ means unpredictable results.

Mean ( $\bar{x}$ ): Find typical values. Calculate average commute time, average spending, or average test scores to understand patterns.
Standard Deviation ( $\sigma$ ): Measure variability. High $\sigma$ in test scores means wide variation; low $\sigma$ means consistent performance.

When working with probability and statistics:

Probability and statistics turn uncertainty into actionable information for better daily decisions!