How do I approach Statistics & Math interview questions?

Statistics & Math questions require understanding of core concepts and practice. PracHub provides solutions with explanations to help you master statistics & math interviews.

What difficulty level is this interview question?

This is a easy difficulty Statistics & Math question, commonly asked during Technical Screen rounds at Snapchat.

What role is this question designed for?

This question is commonly asked for Data Scientist candidates at Snapchat during technical interviews.

Compute expectations and test fairness for coin flips

Q: Compute expectations and test fairness for coin flips

This question evaluates proficiency in probability and statistical inference, specifically expectation and variance of Bernoulli variables, properties of the geometric distribution, and p-value reasoning for hypothesis testing.

You are analyzing repeated flips of a (possibly unfair) coin.

Setup

Let the probability of Heads be $p$ (unknown in general). Assume flips are independent and identically distributed.

Part A — Expected value for an unfair coin

Define a random variable $X$ for a single flip:

$X = 1$ if the flip is Heads
$X = 0$ if the flip is Tails

Compute $\mathbb{E}[X]$ .
(Optional but common follow-up) Compute $\mathrm{Var}(X)$ .

Part B — “Getting a 3” using a geometric distribution

Now flip the coin repeatedly until the first Head appears.

Let $T$ be the number of flips needed to get the first Head (so $T\in\{1,2,3,\dots\}$ ).

Write the distribution of $T$ and identify it.
Compute $\mathbb{P}(T=3)$ in terms of $p$ .
For a fair coin ( $p=0.5$ ), compute the numerical value of $\mathbb{P}(T=3)$ .
Compute $\mathbb{E}[T]$ .

Part C — Is the coin fair? (p-value reasoning)

Suppose you ran this “flip-until-first-Head” experiment once and observed $T=3$ .

You want to test:

$H_0: p=0.5$ (fair coin)
$H_1: p<0.5$ (coin is biased toward Tails; Heads are rarer)

Propose a reasonable p-value for this one observation using an appropriate tail probability under $H_0$ .
Briefly explain what is and is not learnable from a single observation, and what you would do instead to make the test meaningful (e.g., repeat the experiment $n$ times).

Part B — “Getting a 3” using a geometric distribution

Now flip the coin repeatedly until the first Head appears.

Let

T

be the number of flips needed to get the first Head (so

T\in\{1,2,3,\dots\}

Write the distribution of

T

and identify it.

Compute

\mathbb{P}(T=3)

in terms of

p

For a fair coin (

p=0.5

), compute the numerical value of

\mathbb{P}(T=3)

Compute

\mathbb{E}[T]

Part C — Is the coin fair? (p-value reasoning)

Suppose you ran this “flip-until-first-Head” experiment once and observed

T=3

You want to test:

H_0: p=0.5

(fair coin)

H_1: p<0.5

(coin is biased toward Tails; Heads are rarer)

Propose a reasonable p-value for this one observation using an appropriate tail probability under

H_0

Briefly explain what is and is not learnable from a single observation, and what you would do instead to make the test meaningful (e.g., repeat the experiment

n

times).

Compute expectations and test fairness for coin flips

Quick Overview

Setup

Part A — Expected value for an unfair coin

Part B — “Getting a 3” using a geometric distribution

Part C — Is the coin fair? (p-value reasoning)

Solution

Comments (0)

Compute expectations and test fairness for coin flips

Quick Overview

Setup

Part A — Expected value for an unfair coin

Part B — “Getting a 3” using a geometric distribution

Part C — Is the coin fair? (p-value reasoning)

Solution

Comments (0)