Sample Test — Maths and Probability — Product Manager Analytical Interview Preparation

Analytical ability is not just about numbers. It is about understanding the question, breaking down constraints, and applying logic step-by-step.

Talvinder Singh, from a Pragmatic Leaders analytical test session

This set of analytical questions tests your ability to work with probabilities, percentages, growth rates, and combinatorial logic in practical settings. The stakes are real: you must quickly determine whether an opportunity exists within a limited time, understand how growth compounds over months, and calculate revenue changes based on price and volume shifts.

In practice, product managers face similar puzzles — balancing trade-offs, estimating outcomes, and making decisions with incomplete information. Your actual job is to translate the problem into manageable parts and apply the right formulas without losing sight of the context.

The probability of finding an available balloon in 10 minutes

You are at Cappadocia, famous for hot air balloon rides. You want to find a last-minute available spot within 10 minutes. There are 5 balloons lined up in a field. You must check them in sequence, starting with the closest.

Walking times are:

To Balloon 1: 2 minutes
Balloon 1 to Balloon 2: 1 minute
Balloon 2 to Balloon 3: 4 minutes
Balloon 3 to Balloon 4: 2 minutes
Balloon 4 to Balloon 5: 3 minutes

Each balloon has an 85% chance of having an empty spot.

You want to find the probability that you will find an available spot within 10 minutes.

The 10-minute constraint means you can check at most the first 4 balloons because the time to reach the 4th balloon is 9 minutes (2 + 1 + 4 + 2 = 9 minutes), leaving one minute to check.

How to approach this?

Calculate the probability that all first 4 balloons are fully booked — that is, none have an empty spot. If that probability is very low, then the complement is the probability that you find an available spot in one of these balloons.

Probability that a balloon is fully booked = 1 - 0.85 = 0.15
Probability that all first 4 balloons are fully booked = 0.15^4 = 0.000506
Therefore, probability of finding an available spot in any of the first 4 balloons = 1 - 0.000506 = 0.999494 or 99.9%

This means it is almost certain you will find an available balloon within your 10-minute window.

Answer: 99.9% (option d)

Growth rate comparison: When will Red balloons surpass Black balloons?

There are initially 50 Red and 100 Black balloons.

Red balloons grow by 15% per month.
Black balloons grow by 5% per month.

Question: How many months will it take for the number of Red balloons to exceed the number of Black balloons?

Step-by-step:

Each month, multiply the previous count by (1 + growth rate):

Month	Red Balloons	Black Balloons
0	50	100
1	50 * 1.15 = 57.5 ≈ 58	100 * 1.05 = 105
2	58 * 1.15 = 66.7 ≈ 67	105 * 1.05 = 110.25 ≈ 111
3	67 * 1.15 = 77.1 ≈ 78	111 * 1.05 = 116.55 ≈ 117
4	78 * 1.15 = 89.7 ≈ 90	117 * 1.05 = 122.85 ≈ 123
5	90 * 1.15 = 103.5 ≈ 104	123 * 1.05 = 128.9 ≈ 130
6	104 * 1.15 = 119.6 ≈ 120	130 * 1.05 = 136.5 ≈ 137
7	120 * 1.15 = 138	137 * 1.05 = 143.85 ≈ 144
8	138 * 1.15 = 158.7 ≈ 159	144 * 1.05 = 151.2 ≈ 152

At month 8, Red balloons (159) surpass Black balloons (152).

Answer: 8 months (option b)

Revenue impact of price reductions and flight increases

A travel agency operates two balloons, "Birdy" and "Colorful," each charging $15 per trip and making 200 flights per month.

Birdy reduces price by 20%, expects 30% more flights.
Colorful reduces price by 30%, expects 40% more flights.

Calculate total change in revenue.

Initial revenue:

Birdy: 15 * 200 = $3000
Colorful: 15 * 200 = $3000
Total: $6000

After promotion:

Birdy price: 15 * 0.8 = $12
Birdy flights: 200 * 1.3 = 260
Birdy revenue: 12 * 260 = $3120
Colorful price: 15 * 0.7 = $10.50
Colorful flights: 200 * 1.4 = 280
Colorful revenue: 10.5 * 280 = $2940
Total revenue after: $3120 + $2940 = $6060

Change in revenue: 6060 - 6000 = $60 increase

Answer: +60 (option c)

Required price increase to offset flight reduction and increase revenue by 10%

If flights reduce by 10%, by what percentage must the average price increase to ensure total revenue is 10% higher?

Method:

Assume initial trips = 1000, price = 100

Initial revenue = 1000 * 100 = 100,000
Flights after reduction = 1000 * 0.9 = 900
Target revenue = 100,000 * 1.1 = 110,000
Required price = 110,000 / 900 ≈ 122.22
Price increase = 122.22 - 100 = 22.22%

Answer: 22.22% (option d)

Eligibility constraints for hot air balloon pilots

Requirements to work as a pilot:

Valid work permit for Cappadocia region
Valid balloon pilot’s license
Older than 21

In a pool of 100 applicants:

70% do NOT have a valid work permit → only 30% have it
80% have a valid pilot’s license
10% are younger than 21

Maximum possible number of eligible pilots

The maximum is limited by the most restrictive criterion — the smallest percentage meeting a single requirement.

Valid work permit: 30 pilots
Valid license: 80 pilots
Age: 90 pilots

Maximum number of eligible pilots = 30

Answer: 30 (option b)

Minimum number of eligible pilots considering all constraints

Now, find the minimum possible number considering all constraints together.

In the worst case, the 30 pilots with valid work permits could be split into:

20 without a valid license
10 younger than 21

So no pilot satisfies all three constraints simultaneously.

Answer: 0 (option c)

Seating probability: Robert vs Mark

Five people (Robert, Chris, Mark, Scarlett, Jeremy) are taking a balloon ride with 5 seats: 4 fine seats and 1 broken seat.

Robert and Scarlett arrive together.
Mark and Jeremy arrive together.
All have equal probability of arriving in any order.
Each person picks a fine seat if available.
If Robert and Scarlett arrive with only one fine seat, Robert gives it to Scarlett and takes the broken seat.
If Mark and Jeremy arrive with only one fine seat, they flip a coin to decide who gets it.

Question: By what percent is Robert's probability of sitting in a fine seat greater than Mark's?

Approach:

Treat Robert & Scarlett as one unit, Mark & Jeremy as another.
There are 3 groups: (Robert & Scarlett), (Mark & Jeremy), and Chris.
Number of ways to order these groups: 3! = 6
Since Mark & Jeremy can switch seats: multiply by 2 → 12
Since Robert & Scarlett can switch seats: multiply by 2 → 24 total arrangements

Robert and Scarlett arrive last 1/3 of the time → 8 out of 24 arrangements.

In those 8, Robert always takes the broken seat (4 out of 8 times, Scarlett takes broken seat instead).

Subtract these 4 from total → 20 valid seating arrangements.

Robert sits in fine seat in 16 out of 20 arrangements → 80%
Mark sits in fine seat in 12 out of 20 arrangements → 60%

Difference = 20%

Answer: 20% (option c)

Balloon race: distance until Birdy overtakes Colorful

Two balloons fly on the same circular path 500 meters above ground.

Birdy completes a lap in 80 seconds.
Colorful completes a lap in 90 seconds.
They start together.

Question: How many kilometers will Colorful fly until Birdy overtakes it?

Calculation:

Birdy gains 10 seconds per lap (90 - 80).
Birdy needs to gain a full lap (90 seconds) to overtake.
Number of laps needed = 90 / 10 = 9 laps.
Distance Birdy flies = 9 laps * 500 meters = 4500 meters = 4.5 kilometers.

Answer: 4.5 km (option b)

Four workers with different work rates:

Worker	Days to finish alone
Worker1	6
Worker2	8
Worker3	12
Worker4	24

Total payment for building one balloon: ₹5700.

Calculate Worker2's share.

Method:

Find least common multiple (LCM) of days: LCM(6,8,12,24) = 24
Calculate shares by dividing LCM by each worker's days:
- Worker1: 24 / 6 = 4 shares
- Worker2: 24 / 8 = 3 shares
- Worker3: 24 / 12 = 2 shares
- Worker4: 24 / 24 = 1 share
Total shares = 4 + 3 + 2 + 1 = 10
Worker2's share = (3 / 10) * 5700 = ₹1710

Answer: ₹1710 (option c)

Number of balloon registration codes containing "44"

Registration codes are numbers between 1000 and 9999.

Count how many codes contain "44" anywhere:

"44XY"
"X44Y"
"XY44"

Where X and Y are digits (0–9), with constraints:

For "44XY": X and Y can be 0–9 → 10 * 10 = 100 possibilities.
For "X44Y": X cannot be 0 (since codes start from 1000), so X = 1–9 → 9 possibilities; Y = 0–9 → 10 possibilities → 9 * 10 = 90 possibilities.
For "XY44": X = 1–9 → 9 possibilities; Y = 0–9 → 10 possibilities → 9 * 10 = 90 possibilities.

Total = 100 + 90 + 90 = 280 codes.

Answer: 280 (option a)

// exercise: · 10 min

Probability under time constraints

You are at a hot air balloon field in Bangalore with 5 balloons arranged in a line. You have 10 minutes to find an available seat. Walking times between balloons are known, and each balloon has an 85% chance of availability.

How do you calculate the probability that you find a seat within 10 minutes?

Calculate cumulative walking times to each balloon.
Identify how many balloons can be checked within 10 minutes.
Calculate the probability that all these balloons are full.
Subtract from 1 to get the probability of finding a seat.

Hint: Multiply the probabilities of unavailability for each balloon visited within the time limit.

// exercise: · 10 min

Revenue analysis for price and volume changes

A balloon operator reduces ticket prices by 25% but expects a 30% increase in flights.

What is the net effect on revenue?
How would you calculate if revenue increases or decreases?

Hint: Revenue = Price × Number of flights. Calculate new revenue and compare.

// exercise: · 10 min

Eligibility constraint logic

You have 100 pilot applicants.

Requirements:

Work permit: 30% have valid permits.
License: 80% have licenses.
Age: 90% are older than 21.
What is the maximum number of pilots eligible?
What is the minimum number eligible considering all constraints?

Hint: Maximum is limited by the smallest group; minimum assumes maximum overlap of disqualifications.

// exercise: · 10 min

Seating arrangements probability

Five passengers, one broken seat.

Two pairs arrive together.
One pair always gives the fine seat to the other member if only one fine seat is left.
The other pair flips a coin for the fine seat.

Calculate the difference in probabilities that two individuals get a fine seat.

Hint: Use permutations and consider groupings as single units initially.

Where to go next

If you want to practice probability and combinatorics further: Probability and Combinatorics
If you want to strengthen your revenue and growth modeling skills: Financial Modeling for PMs
If you want to improve constraint analysis and logical deduction: Problem Solving Frameworks
If you want to prepare for marketplace and logistics case studies: Marketplace Product Strategy
If you want to practice more analytical test questions: Analytical Interview Preparation

Analytical ability is not just about numbers. It is about understanding the question, breaking down constraints, and applying logic step-by-step.

Talvinder Singh, from a Pragmatic Leaders analytical test session

The probability of finding an available balloon in 10 minutes

Walking times are:

To Balloon 1: 2 minutes
Balloon 1 to Balloon 2: 1 minute
Balloon 2 to Balloon 3: 4 minutes
Balloon 3 to Balloon 4: 2 minutes
Balloon 4 to Balloon 5: 3 minutes

Each balloon has an 85% chance of having an empty spot.

You want to find the probability that you will find an available spot within 10 minutes.

The 10-minute constraint means you can check at most the first 4 balloons because the time to reach the 4th balloon is 9 minutes (2 + 1 + 4 + 2 = 9 minutes), leaving one minute to check.

How to approach this?

Probability that a balloon is fully booked = 1 - 0.85 = 0.15
Probability that all first 4 balloons are fully booked = 0.15^4 = 0.000506
Therefore, probability of finding an available spot in any of the first 4 balloons = 1 - 0.000506 = 0.999494 or 99.9%

This means it is almost certain you will find an available balloon within your 10-minute window.

Answer: 99.9% (option d)

Growth rate comparison: When will Red balloons surpass Black balloons?

There are initially 50 Red and 100 Black balloons.

Red balloons grow by 15% per month.
Black balloons grow by 5% per month.

Question: How many months will it take for the number of Red balloons to exceed the number of Black balloons?

Step-by-step:

Each month, multiply the previous count by (1 + growth rate):

Month	Red Balloons	Black Balloons
0	50	100
1	50 * 1.15 = 57.5 ≈ 58	100 * 1.05 = 105
2	58 * 1.15 = 66.7 ≈ 67	105 * 1.05 = 110.25 ≈ 111
3	67 * 1.15 = 77.1 ≈ 78	111 * 1.05 = 116.55 ≈ 117
4	78 * 1.15 = 89.7 ≈ 90	117 * 1.05 = 122.85 ≈ 123
5	90 * 1.15 = 103.5 ≈ 104	123 * 1.05 = 128.9 ≈ 130
6	104 * 1.15 = 119.6 ≈ 120	130 * 1.05 = 136.5 ≈ 137
7	120 * 1.15 = 138	137 * 1.05 = 143.85 ≈ 144
8	138 * 1.15 = 158.7 ≈ 159	144 * 1.05 = 151.2 ≈ 152

At month 8, Red balloons (159) surpass Black balloons (152).

Answer: 8 months (option b)

Revenue impact of price reductions and flight increases

A travel agency operates two balloons, "Birdy" and "Colorful," each charging $15 per trip and making 200 flights per month.

Birdy reduces price by 20%, expects 30% more flights.
Colorful reduces price by 30%, expects 40% more flights.

Calculate total change in revenue.

Initial revenue:

Birdy: 15 * 200 = $3000
Colorful: 15 * 200 = $3000
Total: $6000

After promotion:

Birdy price: 15 * 0.8 = $12
Birdy flights: 200 * 1.3 = 260
Birdy revenue: 12 * 260 = $3120
Colorful price: 15 * 0.7 = $10.50
Colorful flights: 200 * 1.4 = 280
Colorful revenue: 10.5 * 280 = $2940
Total revenue after: $3120 + $2940 = $6060

Change in revenue: 6060 - 6000 = $60 increase

Answer: +60 (option c)

Required price increase to offset flight reduction and increase revenue by 10%

If flights reduce by 10%, by what percentage must the average price increase to ensure total revenue is 10% higher?

Method:

Assume initial trips = 1000, price = 100

Initial revenue = 1000 * 100 = 100,000
Flights after reduction = 1000 * 0.9 = 900
Target revenue = 100,000 * 1.1 = 110,000
Required price = 110,000 / 900 ≈ 122.22
Price increase = 122.22 - 100 = 22.22%

Answer: 22.22% (option d)

Eligibility constraints for hot air balloon pilots

Requirements to work as a pilot:

Valid work permit for Cappadocia region
Valid balloon pilot’s license
Older than 21

In a pool of 100 applicants:

70% do NOT have a valid work permit → only 30% have it
80% have a valid pilot’s license
10% are younger than 21

Maximum possible number of eligible pilots

The maximum is limited by the most restrictive criterion — the smallest percentage meeting a single requirement.

Valid work permit: 30 pilots
Valid license: 80 pilots
Age: 90 pilots

Maximum number of eligible pilots = 30

Answer: 30 (option b)

Minimum number of eligible pilots considering all constraints

Now, find the minimum possible number considering all constraints together.

In the worst case, the 30 pilots with valid work permits could be split into:

20 without a valid license
10 younger than 21

So no pilot satisfies all three constraints simultaneously.

Answer: 0 (option c)

Seating probability: Robert vs Mark

Five people (Robert, Chris, Mark, Scarlett, Jeremy) are taking a balloon ride with 5 seats: 4 fine seats and 1 broken seat.

Robert and Scarlett arrive together.
Mark and Jeremy arrive together.
All have equal probability of arriving in any order.
Each person picks a fine seat if available.
If Robert and Scarlett arrive with only one fine seat, Robert gives it to Scarlett and takes the broken seat.
If Mark and Jeremy arrive with only one fine seat, they flip a coin to decide who gets it.

Question: By what percent is Robert's probability of sitting in a fine seat greater than Mark's?

Approach:

Treat Robert & Scarlett as one unit, Mark & Jeremy as another.
There are 3 groups: (Robert & Scarlett), (Mark & Jeremy), and Chris.
Number of ways to order these groups: 3! = 6
Since Mark & Jeremy can switch seats: multiply by 2 → 12
Since Robert & Scarlett can switch seats: multiply by 2 → 24 total arrangements

Robert and Scarlett arrive last 1/3 of the time → 8 out of 24 arrangements.

In those 8, Robert always takes the broken seat (4 out of 8 times, Scarlett takes broken seat instead).

Subtract these 4 from total → 20 valid seating arrangements.

Robert sits in fine seat in 16 out of 20 arrangements → 80%
Mark sits in fine seat in 12 out of 20 arrangements → 60%

Difference = 20%

Answer: 20% (option c)

Balloon race: distance until Birdy overtakes Colorful

Two balloons fly on the same circular path 500 meters above ground.

Birdy completes a lap in 80 seconds.
Colorful completes a lap in 90 seconds.
They start together.

Question: How many kilometers will Colorful fly until Birdy overtakes it?

Calculation:

Birdy gains 10 seconds per lap (90 - 80).
Birdy needs to gain a full lap (90 seconds) to overtake.
Number of laps needed = 90 / 10 = 9 laps.
Distance Birdy flies = 9 laps * 500 meters = 4500 meters = 4.5 kilometers.

Answer: 4.5 km (option b)

Four workers with different work rates:

Worker	Days to finish alone
Worker1	6
Worker2	8
Worker3	12
Worker4	24

Total payment for building one balloon: ₹5700.

Calculate Worker2's share.

Method:

Find least common multiple (LCM) of days: LCM(6,8,12,24) = 24
Calculate shares by dividing LCM by each worker's days:
- Worker1: 24 / 6 = 4 shares
- Worker2: 24 / 8 = 3 shares
- Worker3: 24 / 12 = 2 shares
- Worker4: 24 / 24 = 1 share
Total shares = 4 + 3 + 2 + 1 = 10
Worker2's share = (3 / 10) * 5700 = ₹1710

Answer: ₹1710 (option c)

Number of balloon registration codes containing "44"

Registration codes are numbers between 1000 and 9999.

Count how many codes contain "44" anywhere:

"44XY"
"X44Y"
"XY44"

Where X and Y are digits (0–9), with constraints:

For "44XY": X and Y can be 0–9 → 10 * 10 = 100 possibilities.
For "X44Y": X cannot be 0 (since codes start from 1000), so X = 1–9 → 9 possibilities; Y = 0–9 → 10 possibilities → 9 * 10 = 90 possibilities.
For "XY44": X = 1–9 → 9 possibilities; Y = 0–9 → 10 possibilities → 9 * 10 = 90 possibilities.

Total = 100 + 90 + 90 = 280 codes.

Answer: 280 (option a)

// exercise: · 10 min

Probability under time constraints

How do you calculate the probability that you find a seat within 10 minutes?

Calculate cumulative walking times to each balloon.
Identify how many balloons can be checked within 10 minutes.
Calculate the probability that all these balloons are full.
Subtract from 1 to get the probability of finding a seat.

Hint: Multiply the probabilities of unavailability for each balloon visited within the time limit.

// exercise: · 10 min

Revenue analysis for price and volume changes

A balloon operator reduces ticket prices by 25% but expects a 30% increase in flights.

What is the net effect on revenue?
How would you calculate if revenue increases or decreases?

Hint: Revenue = Price × Number of flights. Calculate new revenue and compare.

// exercise: · 10 min

Eligibility constraint logic

You have 100 pilot applicants.

Requirements:

Work permit: 30% have valid permits.
License: 80% have licenses.
Age: 90% are older than 21.
What is the maximum number of pilots eligible?
What is the minimum number eligible considering all constraints?

Hint: Maximum is limited by the smallest group; minimum assumes maximum overlap of disqualifications.

// exercise: · 10 min

Seating arrangements probability

Five passengers, one broken seat.

Two pairs arrive together.
One pair always gives the fine seat to the other member if only one fine seat is left.
The other pair flips a coin for the fine seat.

Calculate the difference in probabilities that two individuals get a fine seat.

Hint: Use permutations and consider groupings as single units initially.

Where to go next

If you want to practice probability and combinatorics further: Probability and Combinatorics
If you want to strengthen your revenue and growth modeling skills: Financial Modeling for PMs
If you want to improve constraint analysis and logical deduction: Problem Solving Frameworks
If you want to prepare for marketplace and logistics case studies: Marketplace Product Strategy
If you want to practice more analytical test questions: Analytical Interview Preparation