Probability
What is the probability of rolling a 5 on a standard die?
1/6 ≈ 0.167 (about 16.7%)
True or False: if you flip heads 5 times in a row, tails is 'due' on the next flip.
FALSE — each flip is independent. The coin has no memory. P(tails) is still exactly 50%.
In the Monty Hall problem: you pick door 1, host reveals a goat behind door 3. You switch to door 2. What is your probability of winning?
2/3 — switching wins 2 out of every 3 times on average.
What is the median of 10, 20, 30, 40, 100?
30
Define "mean" in statistics
The average. Add up all of the values then divide by the number of values.
You flip a fair coin 1,000 times and get 523 heads. Is the coin rigged?
No — this is well within normal variation. 523/1000 = 52.3%, close to 50%. Law of Large Numbers in action.
If you just won a hand of poker, do you have a higher, lower, or the same odds of winning your next hand?
Why does the host's knowledge matter in the Monty Hall problem?
The host never opens the car door — they always reveal a goat. This transfers probability. A random reveal would be different.
What is the mode of 1, 3, 4, 4, 5, 6, 7, 7, 7, 8, 9, 10
7, it is the most common value.
Define the "median" in statistics.
The middle value
A bag has 3 red, 5 blue, 2 green marbles. What is P(blue)?
5/10 = 1/2 = 50%
After hitting 5 shots in a row, is a basketball player more likely to hit the next?
No. Their percentage is almost identical.
If the Monty Hall problem had 100 doors instead of 3, you pick door 1, and the host opens 98 goat doors — should you switch?
Yes. Your original door: 1/100. The remaining door: 99/100. The case for switching gets stronger with more doors.
A company says average employee pay is $90,000. One employee earns $800,000. What would give a better picture of typical pay?
The median.
Define probability.
How likely something is to happen, measured from 0-1 where 0 is impossible and 1 is certain
What is the probability of rolling an even number OR a number less than 3 on a standard die?
Even: {2,4,6}. Less than 3: {1,2}. Combined: {1,2,4,6} = 4 outcomes. P = 4/6 = 2/3
How many people do you need in a room for a 50% chance that two share a birthday?
23 people
What does P(A|B) mean and how does it relate to Monty Hall?
P(A|B) = probability of A given B has occurred. In Monty Hall: P(car behind door 2 | host opens door 3) = 2/3. The host's action is new information that updates the probability.
Name one trick graphs use to make a small change look dramatic.
truncated y-axis (not starting at 0), missing scale, cherry-picked time range, 3D distortion, misleading labels.
What does the Law of Large Numbers state?
As the sample size grows, the sample mean converges to the expected value.