← Bank of America Interview Insights

Bank of America·Software Engineer·Technical Phone Screen·Intermediate

Intermediate
Apr 2026

Summary

Interviewed for a Quant Engineer role at Bank of America and got hit with a probability question that felt deceptively simple at first. The dice problem had two parts and they wanted full derivations, not just answers you'd memorized.

Questions Asked (2)

Q1

You roll two fair six-sided dice. What is the probability that the two dice show different values (i.e., no tie)? Derive it from scratch.

Algorithms & Data StructuresTechnical Trade-offs
Author's notes

I went straight to counting: 36 total outcomes, 6 are ties (1-1, 2-2, etc.), so 30 outcomes where they differ.

Create a free account to read the full note

AI HintsAI Generated

Suggested Approach

Start by clearly defining the sample space for rolling two fair six-sided dice, then compute the probability of a tie and subtract from 1 to get the probability of different values. Alternatively, count the favorable outcomes directly. Explain each step logically to show your reasoning.

Pro tip: Mention that this is a classic probability problem and relate it to real-world scenarios like hashing or load balancing, where avoiding collisions is important. This shows you can connect theoretical concepts to practical engineering problems.

1. Define the sample space

State that each die has 6 equally likely outcomes, so there are 6 × 6 = 36 total possible outcomes when rolling two dice.

2. Identify favorable outcomes

Determine the number of outcomes where the dice show different values. This can be done by counting ties (6 outcomes) and subtracting from 36, or by counting directly: 6 choices for the first die and 5 for the second, giving 30 favorable outcomes.

3. Compute the probability

Divide the number of favorable outcomes by the total number of outcomes: 30/36 = 5/6. Alternatively, compute the probability of a tie (6/36 = 1/6) and subtract from 1: 1 - 1/6 = 5/6.

4. Verify and interpret

Check that the probability is reasonable (close to 1) and explain what it means: there is an 83.33% chance the dice show different values.

Key Points to Mention

  • Sample space size: 36 equally likely outcomes
  • Number of tie outcomes: 6 (one for each value)
  • Probability of tie: 6/36 = 1/6
  • Probability of different values: 1 - 1/6 = 5/6
  • Alternative direct count: 6 × 5 = 30 favorable outcomes
  • Connection to real-world applications like collision avoidance in hashing

AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.

Q2

Given the same two dice, what is the probability that die A specifically shows a strictly higher value than die B? Again, derive it.

Algorithms & Data StructuresTechnical Trade-offs
Author's notes

This is where I fumbled a bit.

Create a free account to read the full note

AI HintsAI Generated

Suggested Approach

First, clarify that the dice are fair and independent, then enumerate all 36 equally likely outcomes. Count the outcomes where die A's value is strictly greater than die B's, and divide by 36 to get the probability.

Pro tip: After computing the probability, mention that by symmetry, the probability that die B is strictly greater than die A is the same, and the remaining outcomes are ties. This shows you understand the structure and can verify your answer.

1. Clarify assumptions

Confirm that both dice are fair and independent, and that each die has 6 faces numbered 1 to 6. State that all 36 ordered pairs (A, B) are equally likely.

2. Enumerate outcomes

List or systematically count the number of outcomes where A > B. For example, for each value of A from 1 to 6, count the number of B values that are strictly less than A.

3. Count favorable outcomes

Sum the counts: when A=1, 0 outcomes; A=2, 1; A=3, 2; A=4, 3; A=5, 4; A=6, 5. Total favorable outcomes = 0+1+2+3+4+5 = 15.

4. Compute probability

Divide the number of favorable outcomes (15) by the total number of equally likely outcomes (36) to get the probability: 15/36 = 5/12 ≈ 0.4167.

5. Verify with symmetry

Note that by symmetry, P(A > B) = P(B > A). The number of ties is 6 (when A=B). So 2 * P(A > B) + 6/36 = 1, leading to P(A > B) = (1 - 1/6)/2 = 5/12, confirming the result.

Key Points to Mention

  • The sample space consists of 36 equally likely ordered pairs (A, B).
  • The event of interest is A > B, not A ≥ B.
  • Counting favorable outcomes by fixing A and counting B values less than A.
  • The sum of arithmetic series 0+1+2+3+4+5 = 15.
  • Probability = favorable outcomes / total outcomes = 15/36 = 5/12.
  • Symmetry argument: P(A > B) = P(B > A) and P(A = B) = 6/36, so P(A > B) = (1 - 1/6)/2 = 5/12.

AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.