Correlation does not imply causation
1 card- 01
Cities with more ice-cream sales have more drownings. What is the most likely explanation?
- A
Eating ice cream causes cramps that lead to drowning
- B
The correlation is a coincidence and would vanish with more data
- C
Drownings make people buy ice cream
- D
A third variable — hot weather — drives both
- A
Monty Hall problem
1 card- 02
A game show has three doors: one hides a car, two hide goats. You pick door 1. The host, who knows where the car is and always opens a goat door, opens door 3. He offers to let you switch to door 2. What should you do?
- A
Stay: doors 1 and 2 are each 1/2, so it makes no difference
- B
Stay: your first pick had 1/3 and nothing has changed for it, but door 2 is also 1/3
- C
Switch, but only because the host is trying to trick you
- D
Switch: door 2 now has a 2/3 chance of the car
- A
Base rate fallacy
1 card- 03
A disease affects 1 in 1,000 people. A test detects it 99% of the time when it is present, and gives a false positive 5% of the time when it is not. A random person tests positive. Roughly how likely is it that they have the disease?
- A
About 99%
- B
About 2%
- C
About 95%
- D
About 50%
- A
Prosecutor's fallacy
1 card- 04
A DNA profile found at a crime scene matches one person in a million by chance. A database of 5 million people is searched and one match is found. Which statement is correct?
- A
The chance that the matched person is guilty is 99.9999%
- B
The chance that the matched person is innocent is one in a million
- C
On the database evidence alone, the match is unremarkable: about 5 innocent matches were expected
- D
A search of a large database makes the match stronger evidence, not weaker
- A
Boy or girl paradox
1 card- 05
A family has two children. You learn that at least one of them is a boy. Assuming boys and girls are equally likely and independent, what is the probability that both are boys?
- A
2/3
- B
1/2
- C
1/3
- D
1/4
- A
Bertrand's box paradox
1 card- 06
Three boxes: one holds two gold coins, one holds two silver coins, one holds one of each. You pick a box at random and draw one coin at random: it is gold. What is the probability that the other coin in that box is also gold?
- A
2/3
- B
3/4
- C
1/3
- D
1/2
- A
Bayes' theorem
1 card- 07
A drawer holds two coins: one fair, one with heads on both sides. You take one at random, flip it twice, and get heads both times. What is the probability that you are holding the two-headed coin?
- A
2/3
- B
3/4
- C
4/5
- D
1/2
- A
Birthday problem
1 card- 08
How many people need to be in a room before it is more likely than not that two of them share a birthday?
- A
23
- B
183
- C
366
- D
57
- A
Complementary event
1 card- 09
You roll a fair die four times. What is the probability of getting at least one six?
- A
Exactly 4/6
- B
About 52%
- C
About 17%
- D
About 67%
- A
Gambler's fallacy
1 card- 10
A fair coin has landed heads five times in a row. What is the probability that the next flip is tails?
- A
More than 1/2: tails is 'due'
- B
1/2
- C
1/64
- D
Less than 1/2: the coin is on a streak
- A
Lottery mathematics
1 card- 11
In a lottery that draws six numbers from 1 to 49, which ticket is more likely to win?
- A
1, 2, 3, 4, 5, 6 — consecutive numbers come up more often than people think
- B
Neither: 1, 2, 3, 4, 5, 6 and 7, 19, 23, 31, 38, 44 have exactly the same chance
- C
7, 19, 23, 31, 38, 44 — sequences are almost never drawn
- D
7, 19, 23, 31, 38, 44 — a random-looking set
- A
Coupon collector's problem
1 card- 12
A cereal box contains one of six toys, each equally likely. On average, how many boxes must you buy to collect all six?
- A
About 11
- B
6
- C
36
- D
About 14.7
- A
Geometric distribution
1 card- 13
You roll a fair die until you get a six. On average, how many rolls does that take?
- A
6
- B
3.5
- C
It has no average: there is no upper limit
- D
About 4
- A
Mutual exclusivity
1 card- 14
Two events each have probability 0.3 and are mutually exclusive. Are they independent?
- A
It cannot be determined without the joint probability
- B
No: if one happens, the other cannot, so knowing one changes the other's probability
- C
Yes, as long as both probabilities are below 0.5
- D
Yes: mutually exclusive events never influence each other
- A
Expected value
1 card- 15
A game costs $10 to play. You roll a die and win $1 per dot shown, plus a $20 bonus on a six. Is the game worth playing, on expected value?
- A
Yes: a six pays $26, which covers the cost with plenty to spare
- B
Yes: the expected winnings are $23.50
- C
It breaks even
- D
No: the expected winnings are about $6.83, less than the $10 cost
- A
St. Petersburg paradox
1 card- 16
A coin is flipped until it lands tails. The prize is $2 if tails comes first, $4 if second, $8 if third, doubling each time. What is the expected prize?
- A
$2
- B
Infinite: every flip adds $1 to the expectation
- C
$4
- D
About $8
- A
Penney's game
1 card- 17
Two players choose a sequence of three coin flips. Alice picks HHT, Bob picks HTH. A fair coin is flipped repeatedly; whoever's sequence appears first wins. Which is true?
- A
It is a fair game because the coin is fair
- B
It is a fair game: both sequences are equally likely on any three flips
- C
Bob is favoured: HTH beats HHT 2 to 1
- D
Alice is favoured: HHT beats HTH 2 to 1
- A
Median
1 card- 18
Five employees earn $40k, $42k, $45k, $48k and $400k. Which single number best describes what a typical employee earns?
- A
The mean, $115k
- B
The median, $45k
- C
The mean, because it uses every value
- D
The mode: there is none, so the data cannot be summarised
- A
Simpson's paradox
1 card- 19
Treatment A has a higher success rate than treatment B for small kidney stones, and also for large ones. Yet overall, B has the higher success rate. How is that possible?
- A
A was given mostly to the harder, large-stone cases, so its overall rate is dragged down by the case mix
- B
One of the two group results must have been miscalculated
- C
The sample sizes were too small for the group results to mean anything
- D
It is not possible: if A wins in every group, it must win overall
- A
Standard error
1 card- 20
A survey of 400 people finds 52% support a proposal, with a margin of error of ±5 percentage points. To halve the margin of error, how many people would need to be surveyed?
- A
About 4,000: ten times the sample
- B
It cannot be halved without changing the confidence level
- C
About 1,600: the margin shrinks with the square root of the sample size
- D
About 800: double the sample, half the margin
- A
Percentage point
1 card- 21
Unemployment rose from 4% to 6%. Which description is correct?
- A
It rose by 50 percentage points
- B
It rose by 2 percentage points, which is a 50% increase
- C
It rose by 2 percentage points, which is a 2% increase
- D
It rose by 2%
- A
Relative risk
1 card- 22
A headline says a food "doubles the risk" of a rare cancer. The baseline lifetime risk is 0.5%. What does the food do to an individual's risk?
- A
Raises it from 0.5% to 1%: an absolute increase of half a percentage point
- B
Raises it by 100 percentage points
- C
Raises it to 50%
- D
Raises it to 2%
- A
Number needed to treat
1 card- 23
In a trial, 4% of patients on placebo had a heart attack within five years, versus 3% on a new drug. How many people must take the drug for five years to prevent one heart attack?
- A
25
- B
100
- C
1
- D
4
- A
Weighted arithmetic mean
1 card- 24
Branch A has 10 employees with an average salary of $50k; branch B has 90 employees averaging $30k. What is the company's average salary?
- A
$35k
- B
$40k
- C
$45k
- D
$32k
- A
68–95–99.7 rule
1 card- 25
Adult heights in a population are roughly normal with mean 170 cm and standard deviation 10 cm. About what fraction of adults are taller than 190 cm?
- A
About 5%
- B
About 16%
- C
About 2.5%
- D
About 32%
- A
Interquartile range
1 card- 26
Which measure of spread is least affected by a single extreme outlier?
- A
The standard deviation
- B
The interquartile range
- C
The variance
- D
The range
- A
Regression toward the mean
1 card- 27
The ten students with the lowest scores on a first test are given a special tutorial. On the second test their average improves markedly. What does this show about the tutorial?
- A
That it works, provided the second test was of equal difficulty
- B
That it works, because the students were selected objectively
- C
That it works: the improvement is measured directly
- D
Little or nothing: the lowest scorers were partly unlucky the first time and would have improved on average anyway
- A
Pearson correlation coefficient
1 card- 28
Two variables have a Pearson correlation of exactly 0. What can you conclude?
- A
There is no linear relationship; a strong non-linear one is still possible
- B
The variables are independent
- C
Knowing one tells you nothing about the other
- D
The data must be noisy
- A
Anscombe's quartet
1 card- 29
Four datasets have the same means, the same variances, the same correlation and the same regression line. What does that tell you about how similar they are?
- A
They differ only in sample size
- B
Almost nothing: Anscombe's quartet has exactly these properties and the four scatter plots look completely different
- C
They are effectively identical
- D
They come from the same underlying distribution
- A
Ecological fallacy
1 card- 30
Countries with higher average chocolate consumption have more Nobel laureates per capita. Which conclusion is justified?
- A
People who eat more chocolate are more likely to win a Nobel prize
- B
Chocolate improves cognition
- C
Nobel laureates eat more chocolate than other people
- D
None about individuals: a relationship between country averages need not hold between people
- A
Berkson's paradox
1 card- 31
Among hospital patients, having disease A appears negatively correlated with having disease B, though the two are unrelated in the general population. Why?
- A
Having A protects against B
- B
The hospital treats A and B in different wards
- C
The correlation must be a statistical fluke
- D
Selection: people are in hospital because they have some disease, so among patients, not having A makes having B more likely
- A
Survivorship bias
1 card- 32
During the Second World War, returning bombers showed the most bullet holes on the wings and fuselage, and few on the engines. Where did statistician Abraham Wald recommend adding armour?
- A
Nowhere: the returning planes proved the existing armour was adequate
- B
Everywhere equally, since the pattern was random
- C
The engines: planes hit there did not return to be counted
- D
The wings and fuselage, where the planes were being hit most
- A
Will Rogers phenomenon
1 card- 33
A hospital reclassifies its borderline cancer patients from "stage 1" to "stage 2" using a more sensitive scan. Afterwards, average survival improves in both stage groups, with no change in treatment. How?
- A
The new scan is itself therapeutic
- B
The moved patients were the sickest of stage 1 and the healthiest of stage 2, raising both averages
- C
The change in averages must be a coincidence
- D
Both groups' survival did not really change; the doctors are misreading the data
- A
Misuse of p-values
1 card- 34
A study reports p = 0.03 for the difference between a drug and placebo. What does that number mean?
- A
There is a 97% chance that the drug works
- B
There is a 3% chance that the drug has no effect
- C
The drug's effect is large
- D
If the drug had no effect, data at least this extreme would occur about 3% of the time
- A
Confidence interval
1 card- 35
A 95% confidence interval for a mean is 10 to 14. Which statement is correct?
- A
There is a 95% probability that the true mean lies between 10 and 14
- B
95% of the data lie between 10 and 14
- C
If the study were repeated many times, about 95% of the intervals computed this way would contain the true mean
- D
95% of future sample means will fall between 10 and 14
- A
Multiple comparisons problem
1 card- 36
A researcher tests 20 independent hypotheses, none of which is actually true, each at the 5% significance level. What is the probability of at least one "significant" result?
- A
100%
- B
About 64%
- C
5%
- D
About 36%
- A
Type I and type II errors
1 card- 37
A smoke detector that almost never goes off when there is no fire, but sometimes fails to go off when there is one, is minimising which kind of error?
- A
Neither: a detector's errors are not statistical
- B
Type I (false positive), at the cost of more type II (false negative)
- C
Type II (false negative), at the cost of more type I
- D
Both at once
- A
Statistical power
1 card- 38
A trial finds no significant difference between two treatments (p = 0.4). What does that show?
- A
That the treatments differ, but not significantly
- B
Only that this trial did not detect a difference; a small study can miss a real effect
- C
That there is a 40% chance the treatments are equal
- D
That the treatments are equally effective
- A
Central limit theorem
1 card- 39
Individual response times on a website are heavily skewed: most are fast, a few are very slow. You repeatedly take random samples of 200 requests and compute each sample's mean. What does the distribution of those sample means look like?
- A
As skewed as the individual response times
- B
It depends on the shape of the individual distribution and cannot be predicted
- C
Approximately normal, centred on the true mean, even though the individual times are skewed
- D
Uniform
- A
Sampling bias
1 card- 40
A newspaper's online poll asks readers whether they prefer print or online news, and 80% of the 20,000 respondents say online. What is the main problem with this result?
- A
The question should have offered more than two options
- B
There is no problem: a sample of 20,000 has a margin of error under 1%
- C
The sample is self-selected from people already reading online; its size does not fix that
- D
20,000 is too few to say anything reliable
- A
Law of large numbers
1 card- 41
A fair coin is flipped 1,000 times and shows 550 heads, 50 more than expected. If it is flipped 1,000,000 more times, what does the law of large numbers predict?
- A
Heads will stay 50 ahead, so the final proportion will be noticeably above 1/2
- B
The next 1,000 flips will show about 450 heads
- C
The proportion of heads will approach 1/2, but the excess of 50 heads is not expected to be cancelled out
- D
Tails will come up more often until the excess is cancelled
- A
Texas sharpshooter fallacy
1 card- 42
An analyst notices that a town has three times the national rate of a rare cancer, and starts investigating the local water supply. What should be checked first?
- A
Whether the water contains any known carcinogen
- B
Whether the town's rate is statistically significant
- C
Whether the cancer is genetic
- D
How many towns were looked at: with thousands of towns, some will show a high rate by chance
- A
Conjunction fallacy
1 card- 43
Linda is 31, single, outspoken and very bright. She majored in philosophy and was deeply concerned with discrimination and social justice. Which is more probable?
- A
Linda is a bank teller
- B
It depends on the base rate of feminists among bank tellers
- C
The two are equally probable
- D
Linda is a bank teller and is active in the feminist movement
- A
Benford's law
1 card- 44
In a large set of naturally occurring financial figures, how often would you expect the leading digit to be 1?
- A
About 11% — one digit in nine
- B
About 50%
- C
About 30%
- D
It depends entirely on the currency
- A
Independence (probability theory)
1 card- 45
Two independent components each fail with probability 0.1 in a year. A system needs both to work. What is the probability the system works through the year?
- A
0.9
- B
0.8
- C
0.81
- D
0.99
- A
End of deck · 45 cards