Sales and Trading Technical Questions: Dice, the Fed and a Trade Pitch

On July 29, 2026, the Federal Reserve left its policy rate where it was, and the Treasury curve still moved two ways at once. Bills and the 2-year yield fell. The 30-year yield rose 11 basis points. A candidate who answers "the Fed held, so not much happened" has missed the only part of the question the interviewer cared about.

That market question sits in the same technical round as the dice game, the double-headed coin and "pitch me a trade". Each is worked below with the arithmetic shown, and every probability answer is reached a second way, because an answer you can reach twice is one you can defend.

The technical round at a glance

Bond math has its own round: price, yield, duration and DV01 are worked in full on bond math interview questions, and a credit desk's judgment questions sit on credit analysis interview questions.

How the rounds are weighted

No bank publishes a split between technical, market and fit questions. What banks do publish is the shape of the process, which shows where each family tends to land.

Candidates fill in the rest. First rounds are described as mostly behavioural: one candidate's first round at a US bank was five pre-set questions, the last of them "pitch a trade or stock". Brain teasers and rapid mental math are described as a trading-seat habit, rare for sales, and the purest probability puzzles cluster at proprietary trading firms more than at the big dealers.

So quick arithmetic and the market question are fair game for anyone, the dice and coin puzzles are most likely on a trading superday, and a pitch can come in any round. The stage-by-stage format is on Sales and Trading Superday, and how a sales interview differs is on Fixed Income Sales.

Expected value: the dice game, worked twice

The classic, as a candidate posted it to a sales and trading forum:

"I can roll a die and collect the amount of money on the die, or if I don't like it, I can roll a second time and I have to pick up the die. What is my expected value?"

The poster said the list came from proprietary trading firms. Start with the rule, not the number. A fresh roll is worth 3.50 on average, so a first roll is worth keeping only if it beats 3.50: keep a 4, 5 or 6, and reroll a 1, 2 or 3.

Method one is the table: the six first rolls are equally likely, so the game is worth their average, 25.5 divided by 6, or 4.25. Method two is the formula: half the time you keep a roll averaging 5, half the time you take a fresh 3.50, and 0.5 × 5 plus 0.5 × 3.50 is 4.25. Enumerating all 36 pairs of rolls under the same rule gives 4.25 a third time.

Then show that you tested the rule. Keep 3s as well, or keep only 5s and 6s, and the game drops to about 4.17 either way: the threshold sits at the value of what you give up. With two rerolls the same logic, worked backward, gives 14/3, about 4.67, keeping only a first 5 or 6. The pay-to-play version is the same question: paid the face of one roll, you would pay anything under 3.50 to play.

Same expected value, different risk

A second question from the same list: a number is drawn at random between 0 and 10 and you are paid that amount. Would you rather it came from the whole real line or only from the integers? The expected value is 5 either way, which is the point. As one reply in the thread put it, "they want to see if you can consider volatility of the outcomes."

The integers 0 to 10 have a variance of 10, from eleven squared minus one, over twelve, and averaging the squared distances from 5 directly gives the same 10. The real line gives 100 divided by 12, about 8.33, which a fine numerical sum reproduces. Same mean, more spread on the integers.

A strong answer gives the number, names the difference in risk, then picks a side with a reason: someone running the bet once and averse to a bad draw takes the real line, while someone paid only on the upside takes the integers.

Conditional probability: seven heads from a jar of quarters

The same list's hardest item reads: "You have 100 quarters in a jar. One of the quarters is double sided (heads). You pick out a random quarter and flip it 7 times, and get all heads. What is the probability you picked the double sided quarter?" Instinct says almost certainly. The prior says otherwise, and the prior is the whole question.

1. Bayes, directly. The trick coin is 1 in 100 and always shows heads. A fair coin shows seven heads 1 time in 128. The chance it is the trick coin is (1/100) divided by (1/100 + 99/100 × 1/128), which is 128/227, about 56.4 percent.
2. Odds, in one line. Out of 12,800 imagined draws, 128 pick the trick coin and all of them show seven heads. The other 12,672 pick a fair coin, and 99 of those show seven heads. That is 128 against 99, the same 128/227.
3. Brute force, as a check. Enumerating all 100 coins against all 128 flip sequences lands on the same fraction.

The follow-up asks for the chance that the eighth flip is heads. Weight the two coins by their new probabilities: 128/227 × 1 plus 99/227 × 1/2 is 355/454, about 78.2 percent. Answering 50 percent treats the coin as fair after seven flips of evidence that it may not be.

Counting questions, and the assumption you say first

"What is the probability of flipping 3 heads and 1 tail?" appears on a university career center's list of the most important sales and trading interview questions, and it has two defensible answers. That is what makes it a good interview question.

If order does not matter, four of the sixteen equally likely sequences contain exactly three heads, because the single tail can sit in any of the four positions. The answer is 4/16, or 25 percent, and the binomial count, four choose three over two to the fourth, gives the same. If the question means one particular order, say heads, heads, heads, tails, the answer is 1/16, or 6.25 percent, which is the figure that list gives.

The same care applies to the classic drill of at least one six in four rolls of a die. The complement, 1 minus (5/6) to the fourth, gives 671/1296, about 51.8 percent, and listing all 1,296 outcomes finds the same 671.

Brain teasers without a probability in them

Not every teaser is about chance. Another item on the same forum list asks how many numbers from 1 to 1,000 can be written as a sum of 4s and 5s.

Write out the small cases: 4, 5, 8, 9, 10, then 12, 13, 14 and 15. Once four consecutive numbers work, adding 4 reaches every number after them, so the only misses are 1, 2, 3, 6, 7 and 11, and the answer is 994.

The check is a formula. For two numbers with no common factor, a and b, exactly (a − 1)(b − 1)/2 positive integers cannot be made, here 3 × 4 / 2 = 6, and the largest is ab − a − b, or 11. Saying the small cases out loud before claiming the rule is the habit being tested.

Mental math: say the method, not only the answer

Mental math is the family most likely to reach any trading seat, and it is scored on the narration as much as the number. Each shortcut below is a sentence you can say while you work.

• Split the awkward number. 47 × 5 is 200 plus 35, so 235. 37 × 7 is 210 plus 49, so 259.
• Round, then repay. 87 + 94 is 87 + 90 + 4, so 181. 845 − 388 is 845 − 400 + 12, so 457.
• Squares near a round number. 49 squared is 50 × 48 + 1, so 2,401. 56 squared is 60 × 52 + 16, so 3,136.
• Difference of squares. 37 × 43 is 40 squared minus 3 squared, so 1,600 − 9, which is 1,591.
• Percent in steps. 7 percent of 350 is 3.5 × 7, so 24.5.
• Square roots by bracketing. The square root of 0.9 sits just under 0.95, because 0.95 squared is 0.9025, so call it 0.949.
• Prices in 32nds. Treasury notes and bonds are quoted in 32nds of a point, so 99-16 is 99 and 16/32, or 99.50, and a plus sign after the 32nds adds a 64th.

On a desk the same skill runs through basis points. A $10 million position in the 10-year note at the July 29, 2026 par yield of 4.67 percent carries a DV01 of about $7,917, so a 6 basis point move costs about $47,500, or roughly 7.92 × 0.06 percent, 0.48 percent, of price. The DV01 arithmetic is on bond math interview questions, and the bond calculator reprices any note, DV01 included, at any yield.

The market question, as candidates hear it

The market family is the one no book prepares you for, because its right answer changes with every Fed meeting. One candidate described the version they got in a fixed income interview:

"I was asked on the December FOMC meeting. What happened there? How did it affect yields?"

They were also asked whether yields rose or fell after the announcement, and why. Interviewers differ on detail: a managing director in the same thread said the ideal answer to where the 10-year sits is a rough round number, so the conversation can move on to what drives rates.

Another interviewer complained that candidates "know their textbook economic theory (economy grows = curve steepens, vice versa), but they don't know current market conditions". A third version asks for both sides: "give me a concise argument for why I should short rates in the near future. And also why I should long rates." The target is the level roughly, the decision exactly, and the reason the curve moved as it did.

A worked answer: the FOMC meeting of July 29, 2026

Take the meeting that ended on July 29, 2026 and answer it in four moves.

The decision, dated and exact. The Federal Open Market Committee held the target range for the federal funds rate at 3-1/2 to 3-3/4 percent, in a statement released at 2:00 p.m. Eastern.

What changed against the last statement. Almost nothing in the words: every sentence about the economy matched June's, including "The Committee will deliver price stability." The vote changed, from 12-0 in June to 9-3, with three members preferring a quarter-point hike, and one small line about reserves was reworded.

3.50-3.75%

target range, held

29 Jul 2026

9-3

the vote

three members wanted a hike

+10bp

2s10s on the day

from 35bp to 45bp

What was priced going in. This is the step most answers skip, and the one that explains the rest.

The move, maturity by maturity. The Treasury strikes its daily par curve from quotes taken at or near 3:30 p.m., after the statement, so the decision-day curve holds the reaction and the day before is a clean baseline.

Bills maturing in four to six months fell most, 10 to 11 basis points, and those span the next two or three meetings. The 2-year fell 4. From the 5-year out yields rose, the 30-year by 11, and the curve steepened 10 basis points on a day the policy rate did not move.

Reading the move by maturity

The decision was a hold, and the 2-year still fell. That is the part that shows understanding: with about one chance in three of a hike priced going in, a hold took some of that probability out, and short maturities rallied as it left.

The long end is harder, and a strong answer says so. One reading is that a committee holding while three members wanted to hike left more inflation risk in long maturities. Offer it as a reading, and say how you would test it, for example by checking whether real yields or inflation compensation drove the move.

A weak answer

• The Fed held, so not much happened
• Yields went up, I think
• The Fed sounded hawkish

A strong answer

• Held at 3-1/2 to 3-3/4 percent on July 29, 2026, by 9 to 3
• Bills and the 2-year fell, the 30-year rose 11, 2s10s steepened 10
• A hike was one-in-three priced, so the front end rallied on the hold

Projections work the same way. At the June 17, 2026 meeting, also a hold, the median projection for the end-2026 rate rose to 3.8 percent from 3.4 in March, and the 2-year rose 15 basis points that day. The July minutes put part of the rise in yields after June down to communications read as more restrictive than expected.

Pitch me a trade: the five parts

A pitch samples the junior job itself: Barclays describes its Markets graduates as "monitoring markets and formulating trade ideas". The advice candidates pass on for rates interviews is to arrive with two or three trade ideas on government bonds and to know the standard curve shapes: steepeners, flatteners and butterflies. Whatever the idea, it needs five parts.

1. The view. One dated sentence with its evidence: which rate or spread moves, and why.
2. The instrument. The cleanest expression of the view. A view on the curve's shape belongs in a curve trade, not an outright position that also bets on the level.
3. The size, in DV01. How much each leg moves per basis point, and why the legs balance.
4. The risk that makes it wrong. The specific move that loses money, ideally with an example of it happening.
5. The exit. A target, a stop and a date. An old rates-interview thread asks exactly this: "how do you pitch an exit price for your interest rate trades?"

The fourth part is the one most pitches skip and interviewers probe: if you cannot argue the other side of your trade, you have not found its risk. The curve mechanics behind steepeners and DV01 hedge ratios are on rates trading.

A worked pitch: a 2s10s steepener after the July meeting

Here is the structure on the curve above, as a candidate might have pitched it on the evening of July 29, 2026: a dated worked example, not a view on where rates go next.

View. The front end had just rallied on a hold that markets had partly priced as a hike, and the long end had sold off. A candidate could argue that a committee reluctant to hike had left more inflation risk in the long end, so 2s10s, at 45 basis points that evening, would steepen further.

Instrument. A DV01-neutral 2s10s steepener: long 2-year notes, short 10-year notes.

Size. Each note is treated as priced at par at that day's par yield.

Both DV01s come out the same two ways, as modified duration times price over 10,000 and by repricing each note a basis point either side of its yield. The 10-year moves about four times as much per basis point, so the 2-year leg is about four times the size.

Risk. A hawkish turn: if the committee signaled or delivered the hike priced for September, the 2-year would sell off harder than the 10-year and the curve would flatten. Six weeks earlier it had: on June 17, 2026 the 2-year rose 15 basis points, the 10-year 6, and 2s10s flattened from 38 to 29.

Exit. A target of 60 basis points, a stop at 35 on closing levels, and a time limit at the September 15-16 meeting.

The risk that makes the pitch wrong, and the exit

The curve went the pitch's way first. By August 17, 2026, 2s10s had widened to 53 basis points, about 8 basis points, or roughly $63,000, in the position's favor. Then the risk case arrived: the 2-year climbed from 4.22 percent on July 29 to 4.63 percent on September 11, the 10-year rose less, and 2s10s closed at 33, through the stop.

Yield by maturity. The maturity axis is on a square-root scale, so every point sits at its true maturity and the short end stays readable.

29 Jul 2026 (the pitch)11 Sep 2026

Curve	1M	3M	6M	1Y	2Y	5Y	10Y	30Y
29 Jul 2026 (the pitch)	3.73%	3.83%	3.97%	4.04%	4.22%	4.37%	4.67%	5.2%
11 Sep 2026	3.93%	4.07%	4.12%	4.35%	4.63%	4.78%	4.96%	5.35%

US Treasury par yields on the evening of the July 2026 FOMC meeting and six weeks later. The 2-year rose 41 basis points and the 10-year 29, so 2s10s flattened from 45 to 33. Source: U.S. Department of the Treasury, Daily Treasury Par Yield Curve Rates.

On closing levels the stop triggered on September 11, a loss of about 12 basis points, roughly $95,000 before carry and financing. The pitch did not fail because the view was silly. It failed exactly as its own risk line described, and the exit, named before the trade began, closed it two basis points past the stop instead of leaving it open.

A daily routine and a view log

The market question rewards a habit more than a cram, and its sources are public and free.

• The Treasury's daily par yield curve. One row a day, 1-month to 30-year, as a downloadable file. Note the 2-year, the 10-year and the gap between them.
• The Fed's statements. The committee scheduled eight meetings for 2026. Read each new statement against the last, sentence by sentence.
• The minutes, about three weeks later. They record what markets had priced going in, which turns a move into an explanation.
• The projections, published at four of those eight. The median policy path is what moved yields in December 2024 and June 2026.

Then keep a view log: one entry a week, five dated lines.

1. The 2-year and 10-year yields and the 2s10s spread, from the Treasury curve.
2. The last Fed decision, its exact range and what changed in the statement.
3. One view, with the evidence behind it.
4. The event that would prove it wrong.
5. The level where you would take profit, and the level where you would cut it.

A month of entries gives the interviewer what they are asking for: a dated view, what happened next and what you learned, which is also the best answer to "pitch me a trade". The other question families are mapped on fixed income interview questions, and a self-test is on the interview readiness check.

The bottom line

Every question in this round is a question about method: the dice game is a decision rule, the double-headed coin a prior, the counting question an assumption and the mental math a narration. The market question is the same test on real data. On July 29, 2026 the answer was a hold at 3-1/2 to 3-3/4 percent, a 9-3 vote, a front-end rally and a long-end selloff, explained by what had been priced.

The pitch built on it needed a view, a DV01 size, a named risk and an exit, and six weeks later the exit was what mattered. Work each question twice before the interview, and say the method out loud when it counts.