Practice: Probability foundations
The goal is fluency with the three rules (complement, addition, multiplication) and the judgment to pick the right one, especially knowing when events are independent. Keep a scratchpad for the computations.
Self-check
Section titled “Self-check”Six short questions. Answer each in your head before opening the collapsible.
1. What is the range of any probability, and what do the endpoints mean?
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Every probability is between 0 and 1. A 0 means the event cannot happen; a 1 means it is certain; 0.5 means as likely as not. Any value reported outside [0, 1] signals a rule was misapplied.
2. When can you compute a probability by counting “favorable over total”?
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Only when the outcomes are equally likely (a fair die, a shuffled deck). Then P(event) is the number of outcomes in the event divided by the total number of outcomes in the sample space. For unequal outcomes you need the rules or data instead.
3. Why is the complement rule the go-to for “at least one”?
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Because “at least one” is awkward to count directly (one, or two, or three…) but its complement, “none,” is usually a single easy product. So P(at least one) = 1 - P(none). One subtraction replaces a messy sum.
4. In the addition rule, why do you subtract P(A and B)?
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Because the outcomes where both A and B happen are counted once in P(A) and again in P(B), so adding double-counts them. Subtracting P(A and B) removes the double count. When A and B cannot both happen, that overlap is zero and the rule is plain addition.
5. What condition must hold to multiply probabilities for an AND, and what goes wrong otherwise?
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The events must be independent, meaning one happening tells you nothing about the other. If they are dependent (one outcome changes the other’s odds), simple multiplication is wrong and you need conditional probability (the next lesson).
6. After four heads in a row on a fair coin, what is the probability the next flip is heads?
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Still 1/2. Independent events have no memory; the coin does not owe you a tails. Believing the next flip is “due” to be tails is the gambler’s fallacy.
Try it yourself: compute the probability
Section titled “Try it yourself: compute the probability”Work each one out, then check. Assume fair dice, fair coins, and a standard 52-card deck.
1. Roll one die. P(odd number)?2. Roll two dice. P(both show a 6)?3. Roll two dice. P(at least one 6)?4. Draw one card. P(a face card [J, Q, K] OR a spade)?5. A pipeline has three independent steps, each succeeding 80% of the time. P(all three succeed)?Show answer
1. P(odd) = {1,3,5} = 3/6 = 1/2.
2. Independent, multiply: P(both 6) = 1/6 x 1/6 = 1/36 (about 2.8%).
3. Use the complement: P(no 6 on a roll) = 5/6, so P(no 6 on either) = 5/6 x 5/6 = 25/36. P(at least one 6) = 1 - 25/36 = 11/36 (about 30.6%).
4. Addition rule, subtract the overlap: P(face card) = 12/52 (J,Q,K in four suits) P(spade) = 13/52 P(face AND spade) = 3/52 (J,Q,K of spades) P(face OR spade) = 12/52 + 13/52 - 3/52 = 22/52 = 11/26 (about 42.3%).
5. Independent, multiply: P(all three) = 0.8 x 0.8 x 0.8 = 0.512 (about 51%).Note item 5: three steps that “almost always work” (80% each) succeed together only about half the time. Chains erode reliability fast.
Try it yourself: which rule, and is it independent?
Section titled “Try it yourself: which rule, and is it independent?”For each, name the rule you would use (complement, addition, or multiplication) and, where it matters, say whether the events are independent.
A. The probability that a single user's request is NOT flagged as spam, given the spam rate is 3%.B. The probability that two different users, acting separately, both complete signup, if each completes 60% of the time.C. The probability that a drawn card is a heart OR a face card.D. The probability of drawing two aces in a row from a deck WITHOUT putting the first card back.Show answer
- A: complement. P(not flagged) = 1 - 0.03 = 0.97.
- B: multiplication, independent. Two separate users act independently, so P(both) = 0.6 x 0.6 = 0.36.
- C: addition. P(heart) + P(face card) - P(heart and face card), subtracting the three heart face cards: 13/52 + 12/52 - 3/52 = 22/52 = 11/26.
- D: NOT simple multiplication, the events are dependent. Without replacement, the first draw changes the odds for the second (after one ace leaves, 3 aces remain in 51 cards). This needs conditional probability, the next lesson. (For the curious: 4/52 x 3/51.)
The discipline: spot “at least one / not” (complement), “or” (addition, subtract overlap), and “and / both” (multiplication, but only multiply directly when independent).
Flashcards
Section titled “Flashcards”Eight cards. Click any card to reveal the answer. Use the Print flashcards button to lay out the full set as one card per page for offline review.
Q. What is the range of a probability, and what do 0 and 1 mean?
Every probability is between 0 and 1. 0 means impossible, 1 means certain, 0.5 means as likely as not. A value outside [0,1] means a rule was misapplied.
Q. Two ways to read a probability?
As a long-run frequency (the fraction a repeated experiment settles toward) or as a degree of belief (a calibrated confidence, like a model’s score). The rules work the same either way.
Q. For equally likely outcomes, how do you compute a probability?
Favorable outcomes divided by total outcomes in the sample space. P(even on a die) = 3/6 = 1/2. Only valid when outcomes are equally likely.
Q. State the complement rule and its main use.
P(not A) = 1 - P(A). Its main use is ‘at least one’: P(at least one) = 1 - P(none), which turns a messy sum into one easy subtraction.
Q. State the addition (OR) rule. Why subtract a term?
P(A or B) = P(A) + P(B) - P(A and B). Subtract P(A and B) because the both-happen cases are counted in each term; otherwise you double-count them. If A and B cannot co-occur, the overlap is 0.
Q. State the multiplication (AND) rule and its condition.
P(A and B) = P(A) x P(B), valid only when A and B are independent (one tells you nothing about the other). For dependent events you need conditional probability.
Q. What does it mean for two events to be independent?
One event happening gives no information about the other (two separate coin flips). Independence is the condition that lets you multiply probabilities directly for an AND.
Q. What is the gambler's fallacy?
Believing past independent outcomes change future ones, like thinking tails is ‘due’ after a run of heads. A fair coin has no memory; each flip stays 1/2.