Probability and Statistics: Normal Distribution, Binomial Probability, and Discrete Distributions

Normal Distribution and Z-Scores

Finding Probabilities Using the Normal Distribution

The normal distribution is a continuous probability distribution that is symmetric about the mean, describing data that clusters around a central value. Z-scores are used to standardize values and find probabilities associated with specific outcomes.

• Mean (μ): The average value in the population.

• Standard Deviation (σ): Measures the spread of the data.

• Z-score Formula: $z = \frac{X - \mu}{\sigma}$ Where X is the value, μ is the mean, and σ is the standard deviation.

• Probability Calculation: Use the Z-score to find the probability from standard normal tables or using the complement rule: $P(Z > z) = 1 - P(Z < z)$

Example:

• Given: μ = 114.8, σ = 13.1, X = 120

• Calculate Z: $z = \frac{120 - 114.8}{13.1} = 0.40$

• Find $P(Z > 0.40)$: $P(Z > 0.40) = 1 - P(Z < 0.40) = 1 - 0.6554 = 0.3446$

Additional info: For sample means, use the standard error: $\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$ For n = 40, $z = \frac{120 - 114.8}{13.1/\sqrt{40}} = 2.51$ $P(Z > 2.51) = 1 - P(Z < 2.51)$

Inverse Normal Problems

Finding Values Corresponding to Given Probabilities

Inverse normal problems involve finding the value (X) that corresponds to a given probability in a normal distribution.

• Given Probability: Find the Z-score that matches the cumulative probability.

• Find X: $X = z \cdot \sigma + \mu$

Example:

• Given: μ = 560, σ = 85, cumulative probability = 0.90

• Z-score for 0.90 is approximately 1.28

• Calculate X: $X = 1.28 \times 85 + 560 = 668.8$

Probability with Tables and Conditional Probability

Using Frequency Tables to Calculate Probabilities

Frequency tables summarize categorical data and allow calculation of probabilities for specific events, including conditional probabilities.

• Marginal Probability: Probability of a single event.

• Conditional Probability: Probability of an event given another event has occurred.

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

Example Table:

	Female	Male	Total
Smoker	38	51	89
Not Smoker	127	10	137
Total	165	61	226

• Probability of selecting a smoker and male: $P(\text{smoker} \cap \text{male}) = \frac{51}{226} = 0.2257$

• Probability of selecting a non-smoker and male: $P(\text{not smoker} \cap \text{male}) = \frac{10}{226} = 0.0442$

• Probability of selecting a non-smoker or male: $P(\text{not smoker or male}) = \frac{204 + 49 + 84}{226} = 1.5044$ Additional info: This sum exceeds 1 due to double-counting; use inclusion-exclusion principle.

Probability Without Replacement

Drawing Cards from a Deck

When drawing cards without replacement, the probability changes after each draw because the total number of cards decreases.

• Probability of Two Face Cards: $P(\text{two face cards}) = \frac{12}{52} \times \frac{11}{51} = 0.04977$

• There are 12 face cards in a standard deck.

Discrete Probability Distributions

Calculating the Mean (Expected Value)

The mean of a discrete probability distribution is the sum of each value multiplied by its probability.

• Formula: $\mu = \sum x_i P(x_i)$

Example Table:

x	P(x)
0	0.40
1	0.31
2	0.23
4	0.02
5	0.04

• Mean: $0(0.40) + 1(0.31) + 2(0.23) + 4(0.02) + 5(0.04) = 0.97$

Binomial Probability

Calculating Binomial Probabilities

The binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success.

• Parameters: n = number of trials, p = probability of success, x = number of successes

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

Example:

• n = 10, p = 0.6, x = 8

• $P(8) = \binom{10}{8} (0.6)^8 (0.4)^2 = 45 \times (0.6)^8 \times (0.4)^2 = 0.1209$