Unit 2: Conditional Probability
What happens once you learn something is true — the sample space shrinks, Bayes' rule flips the conditioning, and independence is stronger than 'unrelated'.
12 topics · 45 questions
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2.1 Defining Conditional Probability
- How it worksHow does conditioning shrink the sample space down to just the outcomes where the condition is true?
- Common mistakeWhy isn't P(A given B) generally equal to P(B given A)?
- Worked exampleFind P(a die roll is greater than 3 given the roll is even) for a fair six-sided die.
- Going furtherWhat happens to P(A given B) when A and B are the exact same event?
Your turn1 of 2
In a group, P(A and B) = 0.154, and P(B) = 0.412. Find P(A given B).2.2 Bayes' Rule
- How it worksHow does Bayes' rule flip P(positive given disease) around into P(disease given positive)?
- Worked exampleFind P(disease given positive) when the disease affects 1 in 1000 people, the test catches 99 percent of cases, and 5 percent of healthy people test positive.
- Common mistakeWhy doesn't P(disease given positive test) equal P(positive test given disease)?
- Ties togetherHow does Bayes' rule connect back to the definition of conditional probability it's built from?
Your turn1 of 2
A disease affects a fraction 0.0351 of the population. A test catches a fraction 0.9584 of true cases (true positive rate) and incorrectly flags a fraction 0.0939 of healthy people (false positive rate). Find P(disease given a positive test).2.2 Bayes' Rule as a Tree of Natural Frequencies
- How it worksHow do natural frequencies replace percentages so a tree's branches can just be counted?
- Worked exampleFind how many of 1000 positive tests are true positives when 10 people truly have the disease and 50 healthy people test positive anyway.
- Going furtherWhat happens to the tree of natural frequencies when the disease becomes ten times more common?
Your turn1 of 2
Out of 1000 people, 36 truly have a disease and all test positive. Among the healthy people, 68 test positive anyway (false positives). Find the fraction of all positive tests that are true positives.2.3 The Law of Total Probability
- How it worksHow does partitioning the sample space let you build P(A) out of conditional pieces?
- Common mistakeWhy doesn't the law of total probability need the partition's pieces to be equally likely?
- Worked exampleFind P(an employee is late) when 60 percent take the train with a 10 percent late rate and 40 percent drive with a 25 percent late rate.
- Going furtherWhat happens to the law of total probability when the partition has 10 pieces instead of 2?
Your turn1 of 2
At a company, a fraction 0.648 of employees take the train, with a late rate of 0.147, and the rest drive, with a late rate of 0.336. Find the overall probability an employee is late.2.4 Independence of Events
- How it worksHow does P(A and B) equaling P(A) times P(B) capture exactly what independence means?
- Worked exampleFind P(a coin lands heads and a die shows 6) when the two events are independent.
- Common mistakeWhy doesn't two events being independent mean they can't both happen?
- Going furtherWhat happens to independence between two events once you condition both on a third event?
Your turn1 of 2
A coin with P(heads) = 0.618 is flipped, and independently, a fair six-sided die is rolled. Find P(the coin lands heads and the die shows a number greater than 3).2.4 Disjoint Events Versus Independent Events
- How it worksHow does knowing one disjoint event happened instantly rule out the other, unlike independence?
- Common mistakeWhy isn't 'independent' just a fancier word for 'disjoint'?
- Going furtherWhat happens to P(A and B) for disjoint events compared with independent events at the same individual probabilities?
Your turn1 of 2
Two independent events A and B each have probability 0.657. Find P(A and B).2.5 The Newton-Pepys Problem
- How it worksHow does the binomial probability formula settle which of the three dice bets wins most often?
- Worked exampleFind the probability of at least one six in 6 rolls, and compare it with at least two sixes in 12 rolls.
- Common mistakeWhy doesn't scaling up the dice count and the required successes by the same factor keep the odds equal?
Your turn1 of 2
Find the probability of at least one six in 15 rolls of a fair die.2.6 Conditional Independence
- How it worksHow does conditioning on a shared cause explain away the correlation between two effects?
- Worked exampleFind whether two symptoms stay independent given a disease, when P(both given disease) equals 0.3 and each alone given disease is 0.6 and 0.5.
- Common mistakeWhy doesn't conditional independence given one event imply the events are independent overall?
- Ties togetherHow does conditional independence connect to the naive-Bayes assumption used in spam filters?
Your turn1 of 2
Two symptoms are conditionally independent given a disease. If P(symptom 1 given disease) = 0.648 and P(symptom 2 given disease) = 0.624, find P(both symptoms given disease).2.7 Simpson's Paradox
- How it worksHow do two scatter groups with opposite within-group slopes combine into one misleading pooled slope?
- Worked exampleFind how Treatment A beats Treatment B on small stones 93 to 87 and large stones 73 to 69, yet trails overall at 78 to 83.
- Common mistakeWhy doesn't a treatment that wins in every subgroup have to win in the combined total too?
- Going furtherWhat happens to Simpson's paradox when the two groups are combined in equal proportions instead of skewed ones?
- Ties togetherHow does Simpson's paradox connect to the difference between P(A given B) and P(A given B and C)?
Your turn1 of 2
Treatment A succeeds at a rate of 0.92 on small-stone patients, who make up a fraction 0.295 of all patients treated, and at a rate of 0.79 on the remaining large-stone patients. Find Treatment A's overall success rate.2.8 The Monty Hall Problem
- How it worksHow does the host's guaranteed goat-reveal carry information that switching cashes in on?
- Common mistakeWhy isn't it true that two remaining doors must each hold a 50-50 chance after one is revealed?
- Worked exampleFind the win rate of switching versus staying, simulated over 300 games of Monty Hall.
- Going furtherWhat happens to the switching advantage when the host opens 98 of 100 doors instead of one of 3?
Your turn1 of 2
A game show has 8 doors, one with a car. You pick one door, and the host (who knows where the car is) opens all but one of the remaining doors, none of which reveal the car. Find the probability of winning the car if you SWITCH to that one remaining door.2.9 Gambler's Ruin
- How it worksHow does first-step analysis turn the ruin probability into a recursive equation you can solve?
- Common mistakeWhy doesn't a slightly unfavorable game just slightly lower the gambler's odds of eventual ruin?
- Worked exampleFind the probability a gambler starting with 20 dollars reaches 100 dollars before going broke, betting 1 dollar at a time on a fair coin.
- Going furtherWhat happens to the gambler's ruin probabilities when the coin is biased instead of fair?
Your turn1 of 2
A gambler starts with 62 dollars and bets 1 dollar at a time on a fair coin, stopping at 0 or at 170 dollars. Find the probability of reaching 170 dollars before going broke.2.9 Random Walks on the Number Line
- How it worksHow does first-step analysis on a random walk mirror the same recursion used in gambler's ruin?
- Worked exampleFind the probability a random walk starting at 0 hits plus 5 before hitting minus 3.
- Going furtherWhat happens to a random walk's position after many steps as the number of steps grows very large?
Your turn1 of 2
A random walk starts at 0 and moves +1 or -1 with equal probability at each step. Find the probability it hits +11 before it hits -10.