Basic Math

Probability | Ninth Grade

Probability - Ninth Grade Math

Introduction to Probability

Probability: The measure of likelihood that an event will occur
Range: $0 \leq P(E) \leq 1$ or 0% to 100%
Experiment: An action or process that produces outcomes
Outcome: A possible result of an experiment
Sample Space (S): The set of all possible outcomes
Event (E): A specific outcome or set of outcomes
Probability Scale:
• $P(E) = 0$: Impossible event (never happens)
• $P(E) = 0.5$: Equally likely (50-50 chance)
• $P(E) = 1$: Certain event (always happens)
• Closer to 1: More likely
• Closer to 0: Less likely

1. Theoretical Probability

Theoretical Probability: Probability based on mathematical reasoning and analysis
Assumes: All outcomes are equally likely
Does NOT require: Actually performing the experiment
Theoretical Probability Formula:

$$P(E) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Possible Outcomes}}$$

Or:
$$P(E) = \frac{n(E)}{n(S)}$$

where:
• $n(E)$ = number of ways event E can occur
• $n(S)$ = total number of possible outcomes in sample space
Example 1: What is the probability of rolling a 4 on a fair die?

Sample space: {1, 2, 3, 4, 5, 6} → 6 possible outcomes
Favorable outcomes: {4} → 1 outcome

$$P(\text{rolling 4}) = \frac{1}{6}$$

Answer: $\frac{1}{6}$ ≈ 0.167 or about 16.7%
Example 2: Probability of drawing a heart from a standard deck of cards

Total cards: 52
Hearts in deck: 13

$$P(\text{heart}) = \frac{13}{52} = \frac{1}{4} = 0.25$$

Answer: $\frac{1}{4}$ or 25%
Example 3: Probability of flipping two heads in a row

Sample space: {HH, HT, TH, TT} → 4 outcomes
Favorable: {HH} → 1 outcome

$$P(\text{two heads}) = \frac{1}{4}$$

Answer: $\frac{1}{4}$ or 25%

2. Experimental Probability

Experimental Probability: Probability based on actual experiments and observations
Also called: Empirical probability
Based on: What actually happened in trials
Requires: Conducting actual experiments
Experimental Probability Formula:

$$P(E) = \frac{\text{Number of Times Event Occurred}}{\text{Total Number of Trials}}$$

Key Point: More trials → more accurate results (closer to theoretical)
Example 1: A coin is flipped 50 times. Heads appeared 28 times.

Experimental probability of heads:
$$P(\text{heads}) = \frac{28}{50} = \frac{14}{25} = 0.56$$

Answer: 0.56 or 56%

Compare to theoretical: $P(\text{heads}) = \frac{1}{2} = 0.50$ or 50%
Note: Experimental is close but not exactly the same
Example 2: A spinner is spun 100 times with these results:

ColorRedBlueGreenYellow
Times32282416

Experimental probabilities:
$P(\text{Red}) = \frac{32}{100} = 0.32$ or 32%
$P(\text{Blue}) = \frac{28}{100} = 0.28$ or 28%
$P(\text{Green}) = \frac{24}{100} = 0.24$ or 24%
$P(\text{Yellow}) = \frac{16}{100} = 0.16$ or 16%

Theoretical vs Experimental Probability

Key Differences:

AspectTheoreticalExperimental
Based onMathematical reasoningActual trials/experiments
FormulaFavorable/Total possibleOccurred/Total trials
RequiresKnowledge of outcomesPerforming experiment
ResultWhat should happenWhat did happen
ExampleP(heads) = 1/2P(heads) = 12/20 in 20 flips

Law of Large Numbers: As number of trials increases, experimental probability approaches theoretical probability

3-4. Probabilities Using Two-Way Frequency Tables

Two-Way Frequency Table: A table showing frequencies of two categorical variables
Also called: Contingency table
Rows: One variable
Columns: Second variable
Cells: Show frequency of each combination
Example 1: Survey of 100 students about favorite subject

MathScienceEnglishTotal
Boys20151045
Girls18221555
Total383725100

Q: What is P(student likes Math)?
$$P(\text{Math}) = \frac{38}{100} = 0.38 \text{ or } 38\%$$

Q: What is P(student is a Girl)?
$$P(\text{Girl}) = \frac{55}{100} = 0.55 \text{ or } 55\%$$

Q: What is P(Girl AND Math)?
$$P(\text{Girl and Math}) = \frac{18}{100} = 0.18 \text{ or } 18\%$$

Conditional Probability with Two-Way Tables

Conditional Probability: Probability of event A given that event B has occurred
Notation: $P(A|B)$ read as "probability of A given B"
Key: The condition limits the sample space
Conditional Probability Formula:

$$P(A|B) = \frac{P(A \text{ and } B)}{P(B)}$$

Or using frequencies:
$$P(A|B) = \frac{\text{Frequency of A and B}}{\text{Frequency of B}}$$

In words: Look only at the row/column of the condition, then find the desired outcome within that row/column
Example 2: Using the table from Example 1

Q: What is P(Math | Boy)? "Probability student likes Math GIVEN student is a Boy"

Method 1 - Direct from table:
Look only at Boy row: 20 like Math out of 45 total boys
$$P(\text{Math}|\text{Boy}) = \frac{20}{45} = \frac{4}{9} \approx 0.44$$

Method 2 - Using formula:
$$P(\text{Math}|\text{Boy}) = \frac{P(\text{Math and Boy})}{P(\text{Boy})} = \frac{20/100}{45/100} = \frac{20}{45} = \frac{4}{9}$$

Answer: $\frac{4}{9}$ ≈ 44.4%
Example 3: From same table

Q: P(Girl | Science)? "Probability student is a Girl GIVEN student likes Science"

Look only at Science column: 22 girls out of 37 total who like Science
$$P(\text{Girl}|\text{Science}) = \frac{22}{37} \approx 0.59$$

Answer: $\frac{22}{37}$ ≈ 59.5%

5. Outcomes of Compound Events

Simple Event: Single outcome
Compound Event: Combination of two or more simple events
Types: AND events, OR events

Ways to Find All Outcomes

Method 1: Tree Diagram
• Shows all possible outcomes branching from each choice
• Good for visualizing compound events

Method 2: List
• Write out all possible combinations systematically

Method 3: Table/Grid
• Create table with outcomes of each event
• Good for two events
Example 1: Flipping a coin and rolling a die

Sample Space using list:
{H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6}

Total outcomes: 2 × 6 = 12

Q: P(Heads and even number)?
Favorable outcomes: {H2, H4, H6} = 3 outcomes
$$P(\text{H and even}) = \frac{3}{12} = \frac{1}{4}$$
Example 2: Choosing outfit: 3 shirts (R, B, G) and 2 pants (J, K)

Tree Diagram outcomes:
R-J, R-K, B-J, B-K, G-J, G-K

Total outcomes: 3 × 2 = 6

6-7. Independent and Dependent Events

Independent Events: The outcome of one event does NOT affect the other
Dependent Events: The outcome of one event DOES affect the other

Identifying Independent vs Dependent

Independent Events Examples:
• Flipping coin twice (first flip doesn't affect second)
• Rolling two dice (one die doesn't affect the other)
• Spinning spinner twice
• Sampling WITH replacement

Dependent Events Examples:
• Drawing cards without replacement (first draw affects second)
• Choosing students without replacement
• Taking items from a bag without putting back
• Sampling WITHOUT replacement
Example 1: Identify if independent or dependent

A: Flip coin, then roll die
→ Independent (coin doesn't affect die)

B: Draw card, keep it, draw another
→ Dependent (first draw changes what's left)

C: Choose marble, replace it, choose again
→ Independent (replacement restores original situation)

D: Choose 2 students from class for a team
→ Dependent (can't choose same person twice)

Probability Formulas

Independent Events:

$$P(A \text{ and } B) = P(A) \times P(B)$$

Multiply the individual probabilities

Dependent Events:

$$P(A \text{ and } B) = P(A) \times P(B|A)$$

where $P(B|A)$ = probability of B after A has occurred

Key: Second probability changes based on first outcome
Example 2: Independent events

Q: Flip coin twice. P(two heads)?

$P(\text{H on first}) = \frac{1}{2}$
$P(\text{H on second}) = \frac{1}{2}$ (independent!)

$$P(\text{HH}) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$$

Answer: $\frac{1}{4}$ or 25%
Example 3: Dependent events

Bag has 5 red and 3 blue marbles. Draw 2 without replacement. P(both red)?

$P(\text{1st red}) = \frac{5}{8}$

After removing 1 red: 4 red, 3 blue, 7 total remain
$P(\text{2nd red}|\text{1st red}) = \frac{4}{7}$

$$P(\text{both red}) = \frac{5}{8} \times \frac{4}{7} = \frac{20}{56} = \frac{5}{14}$$

Answer: $\frac{5}{14}$ ≈ 35.7%

8. Counting Principle

Fundamental Counting Principle: If one event can occur in m ways and another in n ways, then both can occur in m × n ways
Key Word: AND means multiply
Counting Principle Formula:

For multiple events:
$$\text{Total Outcomes} = n_1 \times n_2 \times n_3 \times ... \times n_k$$

where $n_i$ = number of choices for each event

Steps:
1. Count choices for first event
2. Count choices for second event
3. Count choices for each remaining event
4. Multiply all together
Example 1: Restaurant menu

Choose: 1 appetizer (4 choices), 1 main (6 choices), 1 dessert (3 choices)

Total meals possible:
$$4 \times 6 \times 3 = 72$$

Answer: 72 different meal combinations
Example 2: License plate with 2 letters followed by 3 digits

Letters: 26 choices each → $26 \times 26 = 676$
Digits: 10 choices each → $10 \times 10 \times 10 = 1000$

Total plates:
$$26 \times 26 \times 10 \times 10 \times 10 = 676,000$$

Answer: 676,000 possible plates

9-10. Permutations and Combinations

Permutation: Arrangement where ORDER MATTERS
Combination: Selection where ORDER DOES NOT MATTER
Key Question: Does the order of selection matter?

Permutations

Permutation Formula:

Notation: $_nP_r$ or $P(n, r)$
Read as: "n permute r" or "permutations of n things taken r at a time"

$$_nP_r = \frac{n!}{(n-r)!}$$

where:
• $n$ = total number of items
• $r$ = number of items being arranged
• $n!$ (factorial) = $n \times (n-1) \times (n-2) \times ... \times 2 \times 1$

Special Case: Permutation of all n items
$$_nP_n = n!$$
Factorial Values:
$0! = 1$ (by definition)
$1! = 1$
$2! = 2 \times 1 = 2$
$3! = 3 \times 2 \times 1 = 6$
$4! = 4 \times 3 \times 2 \times 1 = 24$
$5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$
Example 1: How many ways can 5 students line up?

Order matters (different positions in line)
This is a permutation of all 5

$$_5P_5 = 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$$

Answer: 120 ways
Example 2: 8 runners in a race. How many ways can 1st, 2nd, 3rd place finish?

Order matters (1st place ≠ 2nd place)
Choose 3 from 8 where order matters

$$_8P_3 = \frac{8!}{(8-3)!} = \frac{8!}{5!} = \frac{8 \times 7 \times 6 \times 5!}{5!} = 8 \times 7 \times 6 = 336$$

Answer: 336 ways

Combinations

Combination Formula:

Notation: $_nC_r$ or $C(n, r)$ or $\binom{n}{r}$
Read as: "n choose r" or "combinations of n things taken r at a time"

$$_nC_r = \frac{n!}{r!(n-r)!}$$

where:
• $n$ = total number of items
• $r$ = number of items being selected

Relationship to Permutations:
$$_nC_r = \frac{_nP_r}{r!}$$

Combinations are always less than or equal to permutations
Example 3: Choose 3 students from 10 for a committee

Order does NOT matter (same 3 people = same committee)
This is a combination

$$_{10}C_3 = \frac{10!}{3!(10-3)!} = \frac{10!}{3! \times 7!}$$

$$= \frac{10 \times 9 \times 8 \times 7!}{3 \times 2 \times 1 \times 7!} = \frac{10 \times 9 \times 8}{6} = \frac{720}{6} = 120$$

Answer: 120 ways
Example 4: Choose 2 toppings from 8 for a pizza

Order doesn't matter (pepperoni + mushroom = mushroom + pepperoni)

$$_8C_2 = \frac{8!}{2!(8-2)!} = \frac{8!}{2! \times 6!} = \frac{8 \times 7}{2 \times 1} = \frac{56}{2} = 28$$

Answer: 28 combinations

Permutation vs Combination Decision Guide

Use PERMUTATION when:
• Order matters
• Different arrangements count as different outcomes
• Keywords: arrange, order, rank, schedule, first/second/third
• Examples: Race finishes, passwords, seating arrangements

Use COMBINATION when:
• Order doesn't matter
• Only the group/selection matters
• Keywords: choose, select, committee, group, team
• Examples: Committee selection, pizza toppings, lottery numbers
Example 5: Decide permutation or combination

A: Select 4 books from 12 to read
→ Order doesn't matter (same 4 books)
→ COMBINATION: $_{12}C_4$

B: Arrange 4 books from 12 on a shelf
→ Order matters (different arrangements)
→ PERMUTATION: $_{12}P_4$

C: Choose 5 students from 20 for a team
→ Order doesn't matter (same team)
→ COMBINATION: $_{20}C_5$

D: Assign 5 students to 5 different roles
→ Order matters (different roles)
→ PERMUTATION: $_{5}P_5 = 5!$

Probability Formulas Summary

TypeFormulaWhen to Use
Theoretical$P(E) = \frac{\text{Favorable}}{\text{Total Possible}}$Mathematical reasoning, equally likely outcomes
Experimental$P(E) = \frac{\text{Occurred}}{\text{Total Trials}}$Based on actual experiment results
Conditional$P(A|B) = \frac{P(A \text{ and } B)}{P(B)}$Probability of A given B occurred
Independent AND$P(A \text{ and } B) = P(A) \times P(B)$Both events, one doesn't affect other
Dependent AND$P(A \text{ and } B) = P(A) \times P(B|A)$Both events, first affects second

Counting Methods Summary

MethodFormulaWhen to UseExample
Counting Principle$n_1 \times n_2 \times ... \times n_k$Multiple choices in sequenceOutfit with 3 shirts and 2 pants: 3×2=6
Permutation$_nP_r = \frac{n!}{(n-r)!}$Arrange r from n, order mattersArrange 3 books from 8: $_{8}P_3 = 336$
Permutation (all)$n!$Arrange all n itemsArrange 5 people: 5! = 120
Combination$_nC_r = \frac{n!}{r!(n-r)!}$Select r from n, order doesn't matterChoose 3 from 8: $_{8}C_3 = 56$

Independent vs Dependent Events

AspectIndependentDependent
DefinitionOutcome of one doesn't affect otherOutcome of one affects the other
Formula$P(A \text{ and } B) = P(A) \times P(B)$$P(A \text{ and } B) = P(A) \times P(B|A)$
ExamplesTwo coin flips, rolling two dice, with replacementWithout replacement, choosing without returning
P(B) changes?NO - stays the sameYES - changes after first event

Permutation vs Combination

AspectPermutationCombination
OrderMATTERSDOESN'T MATTER
Formula$_nP_r = \frac{n!}{(n-r)!}$$_nC_r = \frac{n!}{r!(n-r)!}$
KeywordsArrange, order, schedule, rankChoose, select, group, committee
ExamplesRace places, passwords, seatingTeams, committees, toppings
ResultUsually LARGER numberUsually SMALLER number
Relationship$_nC_r = \frac{_nP_r}{r!}$
Success Tips for Probability:
✓ Probability is always between 0 and 1 (or 0% and 100%)
✓ Theoretical: favorable/possible; Experimental: occurred/trials
✓ For conditional probability, limit to the "given" condition row/column
✓ Independent: outcome doesn't change; Dependent: outcome changes
✓ AND means multiply probabilities
✓ Counting Principle: multiply number of choices
✓ Order matters → Permutation; Order doesn't matter → Combination
✓ Permutations > Combinations for same n and r
✓ Remember: 0! = 1
✓ Practice identifying when to use each formula!
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