Statistics Notes: Probability Distributions, Binomial Distributions, and Uniform Distributions

Section 5.1: Discrete Probability Distributions

When working with probability distributions, it is essential to explicitly write out and check the conditions for any given table or set of data to determine if it qualifies as a legitimate probability distribution.

Conditions for a Probability Distribution

To verify if a distribution is a probability distribution, the following three conditions must be met and documented:

1. Numerical Random Variable $X$: There must be a numerical random variable $X$ associated with a corresponding probability $P(X)$. On a table, this means having numerical values on the left side and probabilities on the right side.
2. Sum of Probabilities: The sum of all probabilities in the distribution must be equal to 1.     $\sum P(X) = 1$
3. Probability Range: Each individual probability $P(X)$ must be between 0 and 1, inclusive.     $0 \le P(X) \le 1$

Example: Probability Distribution for Gender of Children

A couple has three children. Assume boys and girls are equally likely and the gender of one child does not influence the next. Let $X$ represent the number of girls among the three children.

Checking the Conditions:

• Condition 1: The variable $X$ (0, 1, 2, 3) is a numerical random variable associated with probabilities. Status: Met.
• Condition 2: Sum the probabilities.     $\frac{1}{8} + \frac{3}{8} + \frac{3}{8} + \frac{1}{8} = \frac{8}{8} = 1$Status: Met.
• Condition 3: Check if each $P(X)$ is between 0 and 1.
  • $1/8$ is between 0 and 1.
  • $3/8$ is between 0 and 1.     Status: Met.

Conclusion: Since all three conditions are met, this table represents a valid probability distribution.

Section 5.2: Binomial Probability Distributions

A binomial distribution occurs when a specific set of criteria are met regarding trials and outcomes.

Conditions for a Binomial Distribution

1. Fixed Number of Trials: There is a fixed number of trials, denoted as $n$.
2. Independence: The trials must be independent (the outcome of one trial does not affect the others).
3. Two Categories: Each trial has only two possible outcomes, usually categorized as "Success" or "Failure."
4. Constant Probability: The probability of success, denoted as $p$, remains the same in all trials.

Definitions and Variables

• $n$: The number of trials or sample size.
• $x$: The specific number of successes in $n$ trials.
• $p$: The probability of success in a single trial.
• $q$: The probability of failure in a single trial ($q = 1 - p$).
• $P(x)$: The probability of getting exactly $x$ successes.

The Binomial Probability Formula

The standard formula used for calculations is: $P(x) = {}_nC_x \times p^x \times q^{n-x}$

Note: Note that ${}_nC_x$ represents the combination function, often read as "n choose x." An alternative, more complex version of this formula uses factorials: $\\frac{n!}{(n-x)!x!}$. However, using the combination function on a calculator is significantly easier and less prone to parenthetical errors.

Calculator Guidance: TI-83/84 Plus

To find ${}_nC_x$:

1. Enter the value for $n$.
2. Press MATH button.
3. Navigate to PRB or PROB (Probability menu).
4. Select 3: nCr and press ENTER.
5. Enter the value for $x$.

To use the built-in Binomial Function (binomPDF):

1. Press 2nd then VARS (DIST).
2. Scroll to binomPDF.
3. Enter Trials ($n$), $p$, and the $x$ value.
4. Select Paste and ENTER.

Binomial Distribution Wording and Interpretations

The interpretation of probability questions depends heavily on the wording used. The following summary illustrates how to translate verbal phrases into algebraic inequalities for a sample of 4 women asked if reading is their hobby (where $X$ is the number who say yes):

• Exactly 2: $P(X = 2)$
• Less than 2 or Fewer than 2: $P(X < 2) = P(0) + P(1)$
• More than 2: $P(X > 2) = P(3) + P(4)$
• At least 2 or 2 or more: $P(X \ge 2) = P(2) + P(3) + P(4)$
• At most 2 or 2 or less or Up to 2: $P(X \le 2) = P(0) + P(1) + P(2)$

Example 1: Exactly Three Smartphone Users

Problem: Assume 64% of adults with smartphones use them in meetings or classes. If 5 adult smartphone users are randomly selected, find the probability that exactly 3 of them use their smartphones in meetings.

Data Extraction:

• Success: Using smartphone in a meeting.
• $n = 5$ (5 adults selected).
• $x = 3$ (Exactly 3 people).
• $p = 0.64$ (64% success rate).
• $q = 1 - 0.64 = 0.36$.

Formula Setup: $P(3) = {}_5C_3 \times 0.64^3 \times 0.36^{5-3}$$P(3) = 10 \times 0.262144 \times 0.1296$$P(3) = 0.3397$

Conclusion: There is a 33.97% probability that exactly 3 out of 5 adults use their smartphones in meetings.

Example 2: At Least Four Smartphone Users

Problem: 52% of adults with smartphones use them in meetings. If 6 adults are selected, find the probability that at least 4 of them use them.

Data Extraction:

• $n = 6$
• $p = 0.52$
• $q = 1 - 0.52 = 0.48$
• Target: $P(X \ge 4)$

Calculation Steps: We must find $P(4) + P(5) + P(6)$.

1. For $x=4$:     $P(4) = {}_6C_4 \times 0.52^4 \times 0.48^{6-4} \approx 0.2527$
2. For $x=5$:     $P(5) = {}_6C_5 \times 0.52^5 \times 0.48^{6-5} \approx 0.1095$
3. For $x=6$:     $P(6) = {}_6C_6 \times 0.52^6 \times 0.48^{6-6} \approx 0.0198$

Summing the results: $0.2527 + 0.1095 + 0.0198 = 0.3820$

Conclusion: There is a 38.20% chance that at least four people use their smartphones in meetings. It is recommended to use the calculator's memory for intermediate steps to avoid rounding errors.

Significant Values in Probability

In statistics, we use probabilities to determine if a value is "significant" (unusual). The threshold commonly used is 5% ($0.05$).

Rules for Significance:

• Significantly High: If the probability of finding $x$ or more successes is less than or equal to 0.05.     $P(X \ge \text{value}) \le 0.05$
• Significantly Low: If the probability of finding $x$ or fewer successes is less than or equal to 0.05.     $P(X \le \text{value}) \le 0.05$
• Not Significant: If the probability is greater than 0.05 ($P > 0.05$).

Case Study: Believing in Reincarnation

• Poll: 60% of adults believe in reincarnation ($p = 0.6$).
• Sample: 4 adults selected ($n = 4$).
• Question: Is 3 a significantly high number who believe in reincarnation?
• Analysis: To check for "significantly high," find $P(X \ge 3)$.     $P(X \ge 3) = P(3) + P(4) = 0.3456 + 0.1296 = 0.4752$
• Evaluation: $0.4752$ is not less than or equal to $0.05$. In fact, $47.52\% > 5\%$.
• Conclusion: 3 is not a significantly high number of people to believe in reincarnation because the probability of getting 3 or more is greater than 5%.

Section 6.1: Uniform Distributions

A uniform distribution is a continuous distribution where all outcomes within a range are equally likely, forming a rectangular shape on a graph.

Key Concepts of Uniform Distributions

• Total Area: The total area under the probability density curve (the rectangle) is always equal to 1.
• Probability as Area: The probability of a range is equal to the area of the rectangle over that range.
• Rectangle Properties: Area = Length $\times$ Width. In this context, Area = Base $\times$ Height.
• Height Formula: For a distribution from 0 to $k$, the height (probability) is $1 / k$.

Example: Subway Waiting Times

Waiting times for a subway are uniformly distributed between 0 and 5 minutes.

• Base (Range): $5 - 0 = 5$ minutes.
• Height (Probability): Since Total Area = 1, then $1 = 5 \times \text{Height}$. Height = $1 / 5 = 0.2$.

Problem: Find the probability that a passenger waits longer than 2.25 minutes ($P(X > 2.25)$).

1. Identify the shaded region: The range starts at 2.25 and ends at the maximum time of 5.
2. Find the length (base) of the shaded region:     $\text{Length} = 5 - 2.25 = 2.75 \text{ minutes}$
3. Calculate the Area (Probability):     $\text{Area} = \text{Length} \times \text{Height}$$\text{Area} = 2.75 \times (1 / 5)$$\text{Area} = 2.75 \times 0.2 = 0.55$

Conclusion: There is a 0.55 (or 55%) probability that a passenger waits longer than 2.25 minutes.