Probability distributions
Discrete Random Variables
A discrete random variable is a variable whose values are countable or finite, and these values occur randomly. Examples include:
- The number of broken eggs in a carton.
- The number of sixes rolled when throwing four dice.
Characteristics of Discrete Random Variables
- Values are integers (e.g., 0, 1, 2, etc.).
- Each value has a specific probability of occurring.
For example, when flipping two coins, the number of heads that appear is a discrete random variable X, with possible values X ∈ {0, 1, 2}.
Probability Distribution
A probability distribution describes the likelihood of each value of a random variable. For discrete random variables, the probability distribution can be presented as a table, bar graph, or function.
Example: Flipping Two Coins
When flipping two coins, the possible outcomes are:
- 0 heads: both coins show tails.
- 1 head: one coin shows heads and the other tails.
- 2 heads: both coins show heads.
The probabilities for each outcome are:
P(X=0) = 0.25(tails-tails).P(X=1) = 0.5(heads-tails or tails-heads).P(X=2) = 0.25(heads-heads).
x (Number of Heads) |
Probability (P(X=x)) |
|---|---|
| 0 | 0.25 |
| 1 | 0.50 |
| 2 | 0.25 |
The total probability always equals 1:
P(X=0) + P(X=1) + P(X=2) = 0.25 + 0.5 + 0.25 = 1.Applications in Business
In business contexts, probability distributions help companies understand risks and opportunities associated with certain decisions. For example:
If a company plans to launch a new product with a target revenue of at least $50,000 to achieve profitability, but analysis shows there is a 40% chance revenue will fall below $50,000, the company can evaluate this risk before proceeding.
By understanding probability distributions, businesses can make data-driven decisions and prepare strategies to address various possible outcomes.
Probability Distributions for Fair and Unfair Coins
Two Fair Coins
For two fair coins, the probability distribution is:
- P(X=0) = 0.25
- P(X=1) = 0.50
- P(X=2) = 0.25
The bar graph for this distribution would be symmetrical, with the highest bar in the middle (X=1).
Unfair Coin with p=0.4
For p=0.4, the probability distribution becomes:
- P(X=0) = 0.36
- P(X=1) = 0.48
- P(X=2) = 0.16
The bar graph will be skewed to the left, with the highest bar still at X=1.
Unfair Coin with p=0.6
For p=0.6, the probability distribution is:
- P(X=0) = 0.16
- P(X=1) = 0.48
- P(X=2) = 0.36
The bar graph will be skewed to the right, with the highest bar still at X=1.
Comparison of p=0.4 and p=0.6
Both graphs are mirror images of each other. When p<0.5, the graph is skewed left, and when p>0.5, the graph is skewed right.
Pairs of p Values Summing to 1
For example, with p=0.3 and p=0.7:
- When p=0.3, the graph will be more skewed to the left
- When p=0.7, the graph will be more skewed to the right
- Both graphs remain mirror images of each other
Change in P(X=1) as p Increases from 0 to 1
P(X=1) will form a parabolic curve:
- Starting at 0 when p=0
- Reaching a peak of 0.5 when p=0.5
- Returning to 0 when p=1
Meanwhile:
- P(X=0) will decrease linearly from 1 to 0
- P(X=2) will increase linearly from 0 to 1
If plotted on a single graph, these three curves would intersect at p=0.5.
Fermat's Last Theorem
This theorem states that the equation x^n + y^n = z^n has no positive integer solutions for n>2. Although it appears simple, its proof took 350 years. Andrew Wiles finally proved it in 1995 using advanced mathematics unavailable in Fermat's time.
Fermat's correspondence with Blaise Pascal contributed to the development of important concepts in basic probability, namely the idea of equally likely outcomes and expected value.
Expected Value
The expected value or expectation is the average of a discrete random variable, calculated using the formula:
E(X) = Σ xP(X=x)
Example for a biased spinner with probabilities:
- P(X=0) = 0.1
- P(X=1) = 0.3
- P(X=2) = 0.4
- P(X=3) = 0.2
The expected value is 1.7.
Variance Formula
For a discrete random variable X, the variance formula (Var(X)) is:
Var(X) = Σ x²p - {E(X)}²
Where:
- x is the possible value of the random variable X
- p is the probability for each value of x
- E(X) is the expected value of X
- Σ x²p is the sum of squared values multiplied by their probabilities
- {E(X)}² is the square of the expected value
This formula can be remembered as "the average of squares minus the square of the average".
Calculation example using the same spinner probabilities:
- Calculate E(X) = (0×0.1) + (1×0.3) + (2×0.4) + (3×0.2) = 1.7
- Calculate Σ x²p = (0²×0.1) + (1²×0.3) + (2²×0.4) + (3²×0.2)
- Subtract {E(X)}² = 1.7²
The final result is the variance of the discrete random variable X.
