Statistics Study Notes: Probability, Conditional Probability, and Normal Distribution

Probability and Conditional Probability

Basic Probability Concepts

Probability is a measure of the likelihood that an event will occur. It is calculated as the ratio of the number of favorable outcomes to the total number of possible outcomes.

• Probability Formula: $P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$

• Example: If 480 people did not wear a helmet and had head injuries out of a total of 480 + 96 = 576 head injuries, the probability that a randomly selected person with a head injury did not wear a helmet is $\frac{480}{576}$.

Conditional Probability

Conditional probability is the probability of an event occurring given that another event has already occurred.

• Conditional Probability Formula: $P(A|B) = \frac{P(A \cap B)}{P(B)}$

• Example: The probability that a person was not injured, given that they wore a helmet, is calculated by dividing the number of people who wore a helmet and were not injured by the total number of people who wore a helmet.

Contingency Table: Helmet Use and Head Injuries

The table below summarizes the relationship between helmet use and head injuries:

	Head Injuries	NO Head Injuries
Wore Helmet	96	656
Did NOT Wear Helmet	480	2330

Additional info: This table is used to answer probability and conditional probability questions about helmet use and head injuries.

Probability with Multiple Categories

Color Distribution Example

When dealing with probabilities involving multiple categories, such as eye color, the probability of at least one person having a certain characteristic can be found using the complement rule.

• Complement Rule: $P(\text{at least one}) = 1 - P(\text{none})$

• Example: If the distribution of eye color in the U.S. is 40% brown, 35% blue, 12% green, 7% gray, and 6% hazel, the probability that at least one of two randomly selected people has brown eyes is $1 - (0.6 \times 0.6) = 1 - 0.36 = 0.64$.

Expected Value in Decision Making

Expected Value Calculation

Expected value is a key concept in statistics used to determine the average outcome of a random event over many trials.

• Expected Value Formula: $E(X) = \sum [x_i \cdot P(x_i)]$

• Application: In game shows, contestants may use expected value to decide whether to continue playing or take a guaranteed prize.

• Example: If a contestant can win $75, $350, $750, $5000, $50,000, or $1,000,000, the expected value is calculated by multiplying each prize by its probability and summing the results.

Binomial Probability and Worker Firings

Binomial Probability

The binomial probability formula is used to find the probability of a certain number of successes in a fixed number of independent trials.

• Binomial Probability Formula: $P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$

• Application: Used to find the probability that at least four out of five employees were fired due to inability to get along with others, given a known probability.

Mean, Subdivisions, and Probability

Mean and Subdivisions

The mean is the average value of a set of numbers. In the context of city subdivisions, it can be used to analyze the distribution of houses.

• Mean Formula: $\text{Mean} = \frac{\text{Total number of houses}}{\text{Number of subdivisions}}$

• Example: If 535 houses are divided among 576 subdivisions, the mean number of houses per subdivision is $\frac{535}{576}$.

• Probability of Zero Houses: The probability that a subdivision has no houses can be calculated using the binomial or Poisson distribution, depending on context.

Normal Distribution and Heights

Normal Distribution

The normal distribution is a continuous probability distribution that is symmetrical around its mean. It is commonly used to model real-world variables such as heights.

• Normal Distribution Formula: $f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}$

• Z-Score Formula: $z = \frac{x - \mu}{\sigma}$

• Application: Used to find the probability that a randomly selected woman has a height greater than a certain value, or within a certain range.

• Example: If women's heights are normally distributed with a mean of 66.3 inches and a standard deviation of 2.5 inches, the probability that a woman is taller than 71.5 inches can be found using the z-score and standard normal table.

Diagramming Normal Distribution

• Draw a bell-shaped curve centered at the mean (66.3 inches).

• Mark one standard deviation above and below the mean (68.8 and 63.8 inches).

• Shade the area corresponding to the heights of interest (e.g., above 71.5 inches).

Additional info: The questions cover key topics in introductory statistics, including probability, conditional probability, expected value, binomial probability, mean, and normal distribution. These concepts are foundational for understanding data analysis and statistical inference.