seegongsik
Grade 11-12 (age 16-18)

high school Meaning of Probability

Before you roll a die you cannot know which face will come up, yet how likely each one is can be written down exactly as a single number. Probability writes that likelihood as a number from 0 to 1, and when every outcome is equally likely it is the share that the favorable cases take of all the cases. How probabilities add and subtract when events are combined, overlapped or reversed follows from the same idea. Move the slider for the number of elements in event A and the box around A grows along with P(A); switch the Venn diagram slider between union, intersection and complement and the matching formula appears under the figure.

The Intuition of Probability

🎲 What probability is
①It is a number expressing how likely an event is in a trial
②When all outcomes are equally likely, it is the share that the event's cases take of all the cases
③0 ≤ P(A) ≤ 1; an event that must happen has probability 1 and one that cannot happen has probability 0

Sample Space and Events

3
Classical probability
P(A) = n(A)n(S) = |A||S|
When every elementary outcome is equally likely

Operations on Events

union
Addition rule
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
The probability of a union. Adding A and B counts the overlap twice, so subtract it once
Probability of the complement
P(Aᶜ) = 1 − P(A)
The probability that A does not happen

Mutually Exclusive and Independent Events

Mutually exclusive events
A ∩ B = ∅ → P(A ∪ B) = P(A) + P(B)
Two events that cannot happen together. With no overlap, the probabilities simply add
Independent events
P(A ∩ B) = P(A) × P(B)
When whether one event happens does not change the probability of the other
⚠️ Exclusive is not independent
①Mutually exclusive means they cannot happen together (A ∩ B = ∅)
②Independent means neither affects the other's probability (P(A ∩ B) = P(A)P(B))
③If two events with nonzero probability are mutually exclusive, one happening rules out the other, so they are in fact not independent
④A question asking for "at least one" is solved by subtracting its complement, "none at all", from 1

Worked Examples

Example 1
One die is rolled. Find the probability of rolling a multiple of 3.
1
Of the 6 outcomes, the multiples of 3 are 3 and 6, which is 2 cases.
P = 2/6
2
Reduce the fraction.
= 1/3
▸ 1/3
The 6 elementary outcomes are equally likely, so the probability is the ratio of the case counts.
Example 2
One natural number from 1 to 10 is chosen. Find the probability that it is a multiple of 2 or a multiple of 3.
1
Adding the multiples of 2 (5 of them) and the multiples of 3 (3 of them) counts the multiple of 6 (1 of them), which is both, twice.
P(A∪B) = P(A) + P(B) − P(A∩B)
2
Use the addition rule and subtract the 1 overlap.
= 5/10 + 3/10 − 1/10 = 7/10
▸ 7/10
Answering 5 + 3 = 8 counts 6 twice; subtracting the overlap once gives 7.

Summary

Key formulas
P(A) = n(A)n(S), P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
Classical probability and the addition rule
Grade 11 school exam type
Three coins are tossed at once. What is the probability that at least one shows heads?
①1/2
②5/8
③3/4
④7/8
⑤1
▸ ④ 7/8
1
For "at least one", use the complement, "all tails".
P(at least one head) = 1 − P(all tails)
2
The probability of all tails is (1/2)³ = 1/8.
= 1 − (1/2)³ = 1 − 1/8 = 7/8
🎯 Exam Points
①Use P(A) = n(A)/n(S) when all elementary outcomes are equally likely
②The addition rule P(A ∪ B) = P(A) + P(B) − P(A ∩ B) subtracts the overlap once
③Solve "at least" questions with the complement, P(Aᶜ) = 1 − P(A)
④Tell mutually exclusive events apart from independent ones
⑤For independent events, P(A ∩ B) = P(A)P(B)
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Binomial Theorem
Next →
Conditional Probability

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