Rolling a Fair Die

Create a probability distribution for rolling a single fair die.

Identify all possible outcomes: A die can show 1, 2, 3, 4, 5, or 6 = 6 outcomes

Determine probability of each outcome: Fair die means each face is equally likely: $P = \frac{1}{6}$ = $P(x) = \frac{1}{6}$ for each

Verify probabilities sum to 1: $\frac{1}{6} + \frac{1}{6} + \frac{1}{6} + \frac{1}{6} + \frac{1}{6} + \frac{1}{6} = \frac{6}{6} = 1$ = Valid distribution

Write the distribution table: | $x$ | 1 | 2 | 3 | 4 | 5 | 6 | | $P(x)$ | $\frac{1}{6}$ | $\frac{1}{6}$ | $\frac{1}{6}$ | $\frac{1}{6}$ | $\frac{1}{6}$ | $\frac{1}{6}$ | = Uniform distribution

Answer: This is called a **uniform distribution** because all outcomes have equal probability of $\frac{1}{6} \approx 0.167$

Sum of Two Dice

Create a probability distribution for the sum when rolling two dice.

Find all possible sums: Minimum: $1+1=2$, Maximum: $6+6=12$ = Sums range from 2 to 12

Count total outcomes: Die 1 has 6 options, Die 2 has 6 options: $6 \times 6 = 36$ = 36 total outcomes

Count ways to get each sum: Sum 2: (1,1) = 1 way Sum 3: (1,2), (2,1) = 2 ways Sum 4: (1,3), (2,2), (3,1) = 3 ways Sum 7: 6 ways (most common) = Each sum has different frequency

Calculate probabilities: $P(2) = \frac{1}{36}$, $P(3) = \frac{2}{36}$, $P(7) = \frac{6}{36} = \frac{1}{6}$ = Not uniform!

Answer: The distribution is **not uniform** - sum of 7 is most likely ($P = \frac{1}{6}$), while 2 and 12 are least likely ($P = \frac{1}{36}$)

Expected Value Calculation

A game costs 2 euros to play. You win 10 euros if you roll a 6 on a fair die, and nothing otherwise. What is the expected value?

Identify outcomes and payoffs: Roll 6: Win 10 euros, Net profit = 10 - 2 = 8 euros Roll 1-5: Win 0 euros, Net profit = 0 - 2 = -2 euros = Two possible net outcomes: 8 or -2

Find probabilities: $P(\text{roll 6}) = \frac{1}{6}$ $P(\text{roll 1-5}) = \frac{5}{6}$ = Probabilities found

Apply expected value formula: $E(X) = \sum x_i \cdot P(x_i)$ $E(X) = 8 \cdot \frac{1}{6} + (-2) \cdot \frac{5}{6}$ = $E(X) = \frac{8}{6} - \frac{10}{6}$

Calculate: $E(X) = \frac{8-10}{6} = \frac{-2}{6} = -\frac{1}{3}$ = $E(X) \approx -0.33$ euros

Answer: The expected value is $-\frac{1}{3}$ euros (about -33 cents). On average, you lose 33 cents per game - the game favors the house!

Mistake: Probabilities that do not sum to 1

Why: A valid probability distribution MUST have all probabilities sum to exactly 1. If they do not, you have either missed an outcome or made a calculation error.

Correct: Always verify: $\sum P(x_i) = 1$. If the sum is less than 1, you are missing outcomes. If greater, you have counted something twice.

Mistake: Confusing probability with frequency

Why: Frequency is a count (e.g., '6 ways to roll a 7'), while probability is a ratio (e.g., '$P(7) = 6/36$').

Correct: Probability = favorable outcomes divided by total outcomes. Always express as a fraction, decimal, or percentage.

Mistake: Assuming all outcomes are equally likely

Why: Many distributions are NOT uniform. For two dice, sum of 7 is more likely than sum of 2.

Correct: Carefully count the ways to achieve each outcome. Only simple experiments (single die, coin flip) tend to be uniform.

Quality Control in Manufacturing

Factories use probability distributions to predict how many defective items will be produced.

If a machine produces 1000 items and has a 2% defect rate, the expected number of defects is $1000 \times 0.02 = 20$.

Insurance Risk Assessment

Insurance companies use probability distributions to calculate fair premiums based on claim likelihood.

If 5% of drivers file claims averaging 5000 euros, the expected payout per driver is $0.05 \times 5000 = 250$ euros.

Q: What is the difference between discrete and continuous distributions?

A: Discrete distributions have countable outcomes (like dice rolls: 1, 2, 3, 4, 5, 6). Continuous distributions have uncountable outcomes (like height, which can be any value in a range). This lesson focuses on discrete distributions.

Q: Can a probability ever be negative?

A: No! Probabilities are always between 0 and 1 (inclusive). A probability of 0 means impossible, and 1 means certain.

Q: What does expected value tell us?

A: Expected value is the average outcome if you repeated the experiment many times. It does not mean you will get that exact value - it is a long-run average.