Dice & Uniform Problems

Easy2–3 hrsDiceUniformExpected Value

Overview

Dice problems are about recognizing the structure: is this asking for a sum (use linearity), a max/min (use CDF trick), or a sequence until a condition (use Markov/recursion)? The most common mistake is trying to enumerate everything — recognizing the right shortcut is the skill that is tested.

How to Recognize

→Involves rolling dice, spinning wheels, or drawing from a discrete uniform set
→Asked for probability of a sum, maximum, minimum, or specific outcome
→Keywords: 'fair die', 'equally likely', 'roll until you get…'

Step-by-Step Approach

1.Enumerate the sample space for small cases (2 dice → 36 outcomes)
2.Use symmetry — if all outcomes are equally likely, count favorables / total
3.For E[max of k rolls]: use E[max] = Σ P(max ≥ m) = Σ [1 − (m−1/n)^k]
4.For sums, use linearity of expectation — no need to enumerate
5.For 'roll until condition': set up Markov states, write E[x] equations

Key Formulas

E[X] = \frac{n+1}{2}

for a fair

n

-sided die

E[\text{sum of } k \text{ dice}] = \frac{k(n+1)}{2}

P(\max \leq m) = \left(\frac{m}{n}\right)^k

E[\max] = \sum_{m=1}^{n} m \cdot P(\max = m) = \sum_{m=1}^{n} \left[\left(\frac{m}{n}\right)^k - \left(\frac{m-1}{n}\right)^k\right] m

Worked Examples

0Quick Example

Expected max of 2 fair d6: Σ m·[(m/6)²−((m−1)/6)²] for m=1..6 = 161/36 ≈ 4.47.

Problem

Roll two fair 6-sided dice. What is the expected value of the maximum?

Solution

Key insight: P(max ≤ m) = P(both dice ≤ m) = (m/6)² since dice are independent.

P(max = m) = P(max ≤ m) − P(max ≤ m−1) = (m/6)² − ((m−1)/6)²

Expand: this equals (2m−1)/36. Check: Σ(2m−1)/36 for m=1..6 = (1+3+5+7+9+11)/36 = 36/36 ✓

E[max] = Σ m·(2m−1)/36 = (1·1 + 2·3 + 3·5 + 4·7 + 5·9 + 6·11)/36 = (1+6+15+28+45+66)/36 = 161/36 ≈ 4.47

Answer

$E[\max] = 161/36 \approx 4.47$ . The shortcut: $P(\max \leq m) = (m/n)^k$ , then $E[\max] = \sum_m m \cdot \Delta P$ .

Common Mistakes

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Forgetting that dice are independent — P(both show m) = P(one shows m)² only because they're independent.
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Using n+1 formula: E[die] = (n+1)/2 requires n to be the number of faces, not the max face.
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Confusing 'at least one 6' with 'exactly one 6' — use complement for 'at least one': 1 − (5/6)².
!
For E[max of k dice], calculating Σ max × P(max=m) is correct but slow. The CDF shortcut (P(max≤m) = (m/n)^k) is much faster.

Practice Problems

Click "Show Answer" to reveal

You roll 3 fair dice. What is the probability that all three show the same number?

Hint: First die can be anything. Second and third must match it.

Expected number of rolls of a fair die until you see every face at least once (Coupon Collector with n=6).

Hint: When k faces have been seen, expected rolls to see a new face = 6/(6−k).

Roll a fair die. If it shows 1–3, roll again and take the second value. If it shows 4–6, keep it. What is the expected value?

Hint: Use the law of total expectation, condition on the first roll.

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Card & Combinatorics Problems