Worked solutionOne step at a time
Coverage is an average
Coverage c is total letters read divided by genome length. With N reads of length L on a genome of G letters, c = N·L / G. At 1× you read exactly as many letters as the genome has, but that only says how many reads cover a letter on average.
Look at one letter
A read covers a particular letter if it starts in the L positions just before it. Each read starts there with probability L / G, independently of the others.
All N reads must miss it
The chance that every read misses the letter is (1 − L/G)^N. With N = c·G/L and G much larger than L, this is the classic limit e^-c. In words: the number of reads on a letter is Poisson with mean c.
Answer: C, about 37%
At c = 1, e^-1 = 0.368. More than a third of the genome is never read, even though you read as many letters as it has. This is the Lander–Waterman result, and it is why real projects sequence at around 30×: then the fraction missed by chance is e^-30 ≈ 9 × 10⁻¹⁴, well under one letter of a 3.1-billion-letter human genome (in practice the gaps that remain come from regions that are hard to sequence, not from bad luck).
What the computation showedRebuilding a real virus, then rolling the dice
The reel uses the real genome of bacteriophage φX174 (NCBI NC_001422, 5,386 letters, circular) and simulates error-free 150-letter reads at random positions.
- 1,077 reads (30.0× coverage, 983 distinct) were glued by greedy suffix-prefix overlap, longest overlap first, with a minimum of 31 letters. They joined into a single 5,484-letter contig whose two ends overlap by 98 letters; trimming that closes the circle at exactly 5,386 letters, identical to the real genome.
- At 30× the depth per letter ranged from 18 to 45 reads (mean 30). Over 2,000 random 30× trials, no letter of φX174 was ever left uncovered.
- At 1× (36 reads on φX174), 20,000 trials missed on average 36.2% of the genome (spread ±4.0% from trial to trial, because the genome is short).
- At 1× on the first 10 million A/C/G/T letters of human chromosome 21 (66,667 reads), 20 trials missed 36.85% ± 0.07%. The Poisson prediction is e^-1 = 36.79%.
Wrong answersWhy each one is tempting
| Pick | What went wrong |
|---|---|
| A · 0% | Pictured the reads laid end to end. Random reads overlap, so some letters get two or three reads and others get none. |
| B · 5% | Sensed that randomness leaves gaps but underestimated them badly; 5% is what about 3× gives. |
| D · 63% | Computed the right thing and reported its complement: 63% is the part that is read. |
Follow-ups an interviewer asks nextTry these before opening them
What coverage do you need so that at most 1 letter of a 3.1-billion-letter genome is missed by chance?
Solve G·e^-c = 1: c = ln(3.1 × 10⁹) ≈ 21.9. So about 22×. Labs use 30× for margin, because real coverage is uneven (GC-rich and repetitive regions get fewer reads).
At 30×, how many 150-letter reads does a human genome take?
30 × 3.1 × 10⁹ / 150 ≈ 620 million reads. Even 1× is about 20.7 million reads.
Why did the reel's assembly need overlaps of at least 31 letters?
A short overlap can match by chance or inside a repeat, which glues the wrong reads together. 31 letters is long enough to be unique in a small genome. In the human genome, repeats longer than a read (Alu elements, segmental duplications) still break short-read assemblies into pieces, which is why long reads exist.
Greedy overlap or a de Bruijn graph?
Greedy merging is simple and works on a tiny, repeat-poor genome like φX174. Real short-read assemblers split reads into k-mers and walk a de Bruijn graph, which scales to billions of reads and handles repeats and errors more gracefully.
