ML Interview Q Series: Expected Rounds for Random Team Splitting: A Geometric Distribution Analysis
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Six individuals want to randomly divide into two groups of three by simultaneously choosing face-up or face-down hands each round. In any round that yields exactly three face-up and three face-down, they finalize their teams. What is the expected number of rounds needed until such a split occurs?
Comprehensive Explanation
The situation can be understood as a single-round trial where each of the six participants independently chooses either face-up or face-down with equal probability 0.5. We want the event in which exactly three out of the six go face-up, and the remaining three go face-down. The probability of that event occurs in each round, and we keep repeating rounds until it happens for the first time.
Probability of a Successful Split
A successful split is one where three participants choose face-up and the other three choose face-down. Since each person has two choices (face-up or face-down), there are 2^6 total equally likely configurations. Among these, the number of ways to choose exactly three individuals out of six for face-up is n_C_k = 6_C_3 = 20. Therefore, the probability of success in any single round is:
Here, binom(6,3) = 20 is the number of ways to choose which 3 people out of 6 show face-up. Then 2^6 = 64 is the total number of ways the six people can show their hands. Hence, p = 20 / 64 = 5 / 16.
Expected Number of Rounds
Because each round is an independent trial with success probability p, the number of rounds until the first success follows a geometric distribution. For a geometrically distributed random variable X with success probability p, the expected value E(X) = 1 / p. Thus, in this scenario:
Substituting p = 5/16, we get E(rounds) = 16/5 = 3.2 rounds on average. In other words, over many repeated simulations of this process, the long-run average number of rounds needed will be 3.2.
Example Simulation in Python
Below is a small Python snippet that simulates the scenario a large number of times and estimates the average number of rounds required:
import random
import statistics
def one_round_success():
# Each of the six people chooses 'face-up' (1) or 'face-down' (0) with p=0.5
# Returns True if exactly three of them choose 1
choices = [random.randint(0, 1) for _ in range(6)]
return sum(choices) == 3
def simulate(num_trials=10_000_00):
results = []
for _ in range(num_trials):
rounds_count = 0
while True:
rounds_count += 1
if one_round_success():
results.append(rounds_count)
break
return statistics.mean(results)
average_rounds = simulate()
print(f"Estimated average rounds to success: {average_rounds}")
In this simulation:
six random binary choices are made each round.
we check if exactly three choices are 1 (face-up). If so, we count how many rounds that took and record it.
at the end, we compute the mean of the recorded rounds.
The result will get closer to 3.2 as the number of trials increases.
Follow-up Questions
Why does the number of rounds follow a geometric distribution?
In each round, there is a fixed probability p that the event “3 face-up, 3 face-down” occurs. Once a round finishes, it does not affect the next one in any way. Independence between rounds plus a constant probability p of success means the number of trials until the first success follows a geometric distribution. The expectation for a geometric random variable is 1 / p, which justifies the result.
What if the probability of each person choosing face-up is not 0.5?
Then the probability of exactly three face-up depends on p_up ^ 3 * (1 - p_up) ^ 3 multiplied by the binomial coefficient 6_C_3. Formally, in that scenario the success probability in each round becomes:
And we would still use the same geometric distribution logic, leading to an expected round count of 1 / p_new.
Could we have any bias if some individuals coordinate their choices?
The given reasoning assumes independence of choices across people. If some participants try to coordinate or intentionally influence others, the assumption that each trial is identically and independently distributed may no longer hold. Real-world biases, such as peer influence or patterns in choice, would change the probability of obtaining three face-up and three face-down in each round. The analysis here is strictly for the idealized scenario where each person’s choice is independent and equally likely.
What about a scenario with a different total number of people?
If there are 2n people in total, we want n people choosing face-up and n people choosing face-down. The probability in a single round becomes:
And the expected number of rounds is 1 / p_{2n}. However, for large 2n, the binomial coefficient can become quite large, so the probability will vary, and the expected number of rounds might change significantly.
Are there potential issues in implementing the procedure with a small or large group?
For a small group (like 2 people), the problem becomes trivial because the only way to split equally is if each person chooses differently. For a large group, the main concern is ensuring genuine randomness in each individual’s choice and not inadvertently skewing the probability. Furthermore, from a practical standpoint, verifying 3 face-up vs. face-down in bigger groups might require more sophisticated means of counting or checking.
These considerations highlight how the core mathematical approach is quite robust, but real-life issues—such as correlation in choices or bias—can alter the theoretical expectation.
Below are additional follow-up questions
What if the random generator or selection process is flawed and not truly random?
If the method used to generate each individual’s face-up or face-down choice is biased or exhibits predictable patterns, the assumption that each trial has the same probability p of success may break down. For example, if some random generator tends to produce more "face-up" results than "face-down," the probability of exactly three face-up choices changes each round. In practice:
Subtle biases can arise from flawed random number generation algorithms, particularly if the seed is deterministic or repeated.
Human bias might creep in if participants prefer face-up out of habit.
Over many rounds, if participants observe patterns or develop superstitions, the probability of obtaining a 3–3 split might deviate from 5/16 each trial.
Because the expected value derivation depends on a constant probability of success each round, any systematic deviations from that assumption could lead to a longer or shorter average waiting time. Organizations might mitigate this risk by using well-established cryptographic random generators or physically verifiable randomization methods (like coin flips or dice rolls).
How might correlated choices among participants affect the outcome?
If participants’ choices are not made independently, the probability that exactly three select face-up is no longer 6_C_3 / 2^6 in each round. Correlation can arise when:
A pair (or group) of individuals consistently coordinate their choices (knowingly or unknowingly).
Social influence, peer pressure, or shared preferences cause multiple participants to show face-up or face-down together.
When correlation is present:
The distribution of (number of face-up) might shift. For instance, a cluster of five people might choose face-up together frequently.
The probability p of success per round changes, invalidating the straightforward geometric approach.
One might need to model the joint probability structure (a multivariate distribution that accounts for correlation) to calculate the correct success probability each round.
If correlations are strong, we might find that 3–3 splits happen less or more frequently than in the independent scenario. The expected number of rounds must then be derived from the actual distribution or estimated empirically.
Could the teams end up being the same people repeatedly if they try multiple rounds?
When the event (exactly three face-up, three face-down) occurs, we get a valid split, but one might worry about identical teams reoccurring across multiple attempts if the same individuals tend to choose face-up. This is less about the probability of getting a 3–3 split and more about the composition of the teams:
Even though the probability of a 3–3 split is 5/16 each round, any particular arrangement of who’s face-up and who’s face-down within that split can reappear.
In truly independent repeated attempts, the chance of repeating the same grouping is still possible but typically small.
If participants unconsciously repeat their past choices (or coordinate in the same manner), the same team grouping might recur, defeating the goal of “randomness” in team composition.
To mitigate this risk, one might ask each participant to randomize their choice differently each round or use external randomness (like throwing dice) to ensure variation.
How do we account for the variance in the number of rounds until a successful split occurs?
The number of rounds until the first 3–3 split follows a geometric distribution under independence with success probability p = 5/16. For a geometric random variable, the variance is:
Var(X) = (1 – p) / p^2 in plain text notation.
Specifically, with p = 5/16:
1 – p = 11/16,
p^2 = (5/16)^2 = 25/256, so Var(X) = (11/16) / (25/256) = (11/16) * (256/25) = 176/25 = 7.04 in decimal form.
In practice, this variance means that while on average you expect 3.2 rounds, you can have many trials that take 1 round (if you’re lucky) or 10 or more rounds (if you’re unlucky). Realistically, team formation might require patience because of this spread. It’s also useful to note that if p changes over time due to biases or other external factors, the actual variance of the waiting time might differ from the theoretical calculation.
How do we handle a situation where the process might be stopped prematurely?
Sometimes, time constraints might force the group to stop after a certain number of rounds, even if they haven’t yet achieved a 3–3 split. For instance, if the group only allows five attempts before they must move on:
The probability of achieving at least one success within five attempts can be computed as 1 – (1 – p)^5.
If they fail to get a 3–3 split in those five rounds, they might resort to an alternative method like forcibly assigning teams or flipping a single coin to decide a final arrangement.
The expectation of the number of rounds until success is theoretical for an unbounded process; if the process is capped, the final result is no longer purely geometric.
In real-life team setups, imposing a maximum number of attempts is quite common. People might not want to endlessly cycle through random attempts without a fallback plan.
Are there any alternative methods to ensure an equal split without waiting for a random 3–3 configuration?
Relying on pure chance in each round might be too uncertain for some applications. Alternative methods include:
Drawing names from a hat or using a random permutation of the six people, then splitting the first three into one team and the next three into another. This ensures an immediate, random 3–3 grouping.
Using random pairing algorithms that systematically track who has been teamed together and ensure different matchups.
Algorithmic solutions where each person is randomly assigned a number from 1 to 6, and people with ranks 1–3 form one team, and 4–6 form the other.
These approaches guarantee one-shot formation but lose the “fun” factor of flipping hands to decide. In a real-world scenario, if the group needs a quick solution, systematic randomization is typically preferred to repeated trials with an uncertain stopping time.
Could participants exploit knowledge of partial outcomes (like seeing a neighbor’s choice) before deciding?
If the rules say everyone’s choice is revealed simultaneously, in principle no one should know the others’ selections in advance. But in practice, if someone glimpses a neighbor’s hand or if participants reveal their decisions at slightly staggered times:
The probability of exactly three face-up vs. face-down might shift.
Someone seeing multiple face-up hands might deliberately choose face-down to force a 3–3 split or vice versa.
The process can become strategic, introducing game-theoretic elements. If participants aim to speed up or delay the formation of teams, they might coordinate or counteract each other’s choices.
This undermines the assumption of independence. Modeling such a scenario would require analyzing it as a game rather than simple independent Bernoulli choices. Strategies, partial information, and rational (or irrational) player behavior can drastically alter the probability distribution and expected waiting time.