Probability from Equally Likely Outcomes

What you’ll learn

Count a well-defined sample space and calculate simple probability.

រាប់លទ្ធផលអាចកើត និងគណនាប្រូបាបសាមញ្ញ។

Meaning and use

For equally likely outcomes, probability is the number that satisfy the event divided by the total. Equally likely is an assumption, not a consequence of listing outcomes. A probability lies between zero and one. The complement is the event not happening.

បើលទ្ធផលនីមួយៗមានឱកាសស្មើគ្នា ប្រូបាបស្មើចំនួនលទ្ធផលអំណោយផលចែកចំនួនសរុប។ មិនអាចប្រើច្បាប់រាប់នេះដោយមិនពិនិត្យឱកាសស្មើគ្នាទេ។

Formulas

Equally likely outcomes

\[P(E)=\frac{n(E)}{n(S)}\]

Complement

\[P(\text{not }E)=1-P(E)\]

Symbols and units

Symbol	Meaning	Unit or restriction
S	Sample space	Set of possible outcomes
E	Event of interest	Subset of S
n(S), n(E)	Counts of outcomes	Nonnegative integer counts; n(S) > 0
P(E)	Probability	Dimensionless, from 0 to 1

Conditions and limits

• The counting formula requires equally likely elementary outcomes.
• Count distinct possible outcomes, not merely the names of broad categories.
• ប្រូបាប 1/2 មិនធានាថាលទ្ធផលពីរដងនឹងកើតម្ដងម្នាក់ទេ។

Worked examples

A fair die

Find the probability of an even number on a fair six-sided die.

1. Sample space has 6 outcomes.
2. Even outcomes are 2, 4, 6: three outcomes.
3. Calculate \(P(E)=3/6=1/2\).

\(\frac12\)

រាប់លទ្ធផលគូ 3 ក្នុងចំណោម 6 ដែលមានឱកាសស្មើគ្នា។

A bag of counters

A bag has 3 red and 2 blue counters, selected uniformly. Find the probability of not red.

1. Total \(3+2=5\).
2. Not red means blue, so \(P=2/5\).

\(\frac25\)

ករណីមិនក្រហមនៅទីនេះជាពណ៌ខៀវ។

Common mistakes

Instead of	Use	Why
A probability can be 1.5.	A probability cannot exceed 1.	Check the outcome count and assumptions.
Two named results always have probability 1/2 each.	They do only if equally likely.	The names alone do not establish equal chances.

Practice

Solve these before revealing the worked solutions. Include units where relevant.

សាកល្បងដោះស្រាយមុនបើកមើលចម្លើយ។ សូមបង្ហាញជំហាន និងឯកតាដែលពាក់ព័ន្ធ។

1. A fair die is rolled. Find P(rolling 1).
2. If P(E) = 0.3, find P(not E).
3. A bag has 4 red and 6 blue counters selected uniformly. Find P(red).

Answers and steps

Show solutions / បង្ហាញដំណោះស្រាយ

1. 1/6

   One favourable outcome out of six equally likely outcomes.

2. 0.7

   1 − 0.3 = 0.7.

3. 2/5

   4/10 simplifies to 2/5.

Quick recap

• Define the sample space.
• Check equal likelihood before counting.
• A probability is between 0 and 1.