THE GUT CHECK · VOLUME 1
Can you trust your gut?
Eight questions. A few surprises. A better feel for the odds.
From coin flips to strange coincidences, see where your instincts get it right and where the numbers have other ideas.
8 questions · About 3 minutes · No signup
No timer. No trick wording. Every answer comes with an explanation.
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A fair coin has landed heads five times in a row. Each toss is independent. What is the chance the next toss is tails?
- More than 50%
- Exactly 50%
- Less than 50%
Correct answer: Exactly 50%
Tails isn't “due.” The next toss still has a 50% chance of landing tails, regardless of the five tosses before it. A streak changes the story in your head, not the coin's next toss.
Show the math
Independence means P(tails next | five heads) = P(tails next) = 1/2.
You roll two fair, independent six-sided dice. Which total is more likely?
- A total of 6
- A total of 7
- They are equally likely
Correct answer: A total of 7
There are six ways to roll a total of 7, but only five ways to roll a total of 6. The totals look like two equal choices; the combinations underneath them aren't equal.
Show the math
The 36 ordered pairs are equally likely. For 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1), so 6/36 ≈ 16.7%. For 6: (1,5), (2,4), (3,3), (4,2), (5,1), so 5/36 ≈ 13.9%.
Before six independent tosses of a fair coin, you choose one exact sequence. Which sequence is more likely to occur?
- H H H H H H
- H T T H T H
- They are equally likely
Correct answer: They are equally likely
Each exact six-toss sequence has a 1 in 64 chance. The mixed sequence looks more random, but that doesn't make this particular sequence more likely than six heads. All mixed sequences combined are a different, much larger group of outcomes.
Show the math
Each specified result contributes a factor of 1/2. (1/2)^6 = 1/64 = 1.5625% for either exact sequence.
In a simplified forecast, rain has a 50% chance on Saturday and a 50% chance on Sunday. Assume the two days are independent. What is the chance it rains on at least one day?
- 50%
- 100%
- 75%
Correct answer: 75%
There are four equally likely possibilities: dry/dry, rain/dry, dry/rain, and rain/rain. Three include rain. Actual weather on consecutive days may be linked; the 75% answer depends on this question's independence assumption.
Show the math
P(at least one rainy day) = 1 − P(both dry) = 1 − (0.5 × 0.5) = 0.75.
A bag contains three red tokens and one blue token. You randomly draw a red token and leave it out. What is the chance the next random draw is blue?
- 1 in 4
- 1 in 3
- 1 in 2
Correct answer: 1 in 3
Removing the red token leaves three tokens: two red and one blue. Unlike independent coin tosses, the first draw changes what is available for the next draw.
Show the math
After removing a red token, P(blue next) = 1 blue token / 3 remaining tokens = 1/3 ≈ 33.3%.
In a hypothetical game, a rule change lowers the chance of losing from 2 in 100 to 1 in 100 per play. Which statement is accurate?
- The chance fell by 50% relative to its original level
- The chance fell by 1 percentage point
- Both statements
Correct answer: Both statements
Going from 2% to 1% cuts the chance in half, a 50% relative reduction. It also lowers the chance by one percentage point. Both are true, but they can sound very different without the starting number.
Show the math
Relative reduction = (2% − 1%) / 2% = 50%. Absolute change = 2% − 1% = 1 percentage point.
Each independent attempt in a game has a 1 in 10 chance of success. With 10 attempts, what is the chance of at least one success?
- About 65%
- 100%
- 10%
Correct answer: About 65%
Repeated attempts improve your chances, but they don't guarantee a win. Missing all ten times has about a 35% chance, leaving about a 65% chance of at least one success.
Show the math
P(no success) = 0.9^10 ≈ 0.348678. P(at least one success) = 1 − 0.9^10 ≈ 0.651322, or 65.1%.
Imagine 23 people whose birthdays are independent and equally likely to fall on any of 365 days. Ignore leap years. What is the chance that at least two share a birthday?
- About 6%
- About 25%
- About 51%
Correct answer: About 51%
In a group of 23, there are 253 different pairs of people who could match. We're asking whether any pair shares a birthday, not whether someone shares yours. Under this simplified model, the chance is just over half. Real birthdays are not perfectly uniform.
Show the math
P(no shared birthdays) = (365/365) × (364/365) × … × (343/365) ≈ 0.492703. Subtract from 1 to get approximately 50.7%. The 253 pair comparisons are not mutually independent, so multiplying a single-pair chance by 253 does not give the exact answer.