Probability from Data

For Elementary Students

Two Types of Probability

There are two ways to think about probability:

1. What SHOULD happen (Theoretical)

• A coin should land on heads half the time
• You figure this out with math

2. What ACTUALLY happens (Experimental)

• When you flip a coin 10 times, you might get 6 heads
• You figure this out by collecting data

Think about it like this: Theoretical is like a prediction, experimental is like checking to see what really happened!

Experimental Probability

Experimental probability uses real data you collect!

Formula:

Experimental Probability = Times it happened ÷ Total tries

Example: You roll a die 20 times. You get a 3 exactly 4 times.

• Times you got a 3: 4
• Total rolls: 20
• Probability: 4 ÷ 20 = 1/5 or 20%

Frequency Tables

A frequency table is like a tally chart — it counts how many times each thing happens!

Example: You ask 15 friends about their favorite sport:

Finding probability from the table:

What's the probability a random friend likes baseball?

• Baseball frequency: 7
• Total friends: 15
• Probability: 7/15 ≈ 47%

Making a Frequency Table

Steps to make your own:

Step 1: Collect your data

Colors drawn from a bag:
Red, Blue, Red, Green, Red, Blue, Red, Green, Blue, Red

Step 2: Count each type (make tallies)

Step 3: Find probabilities

• P(Red) = 5/10 = 1/2 = 50%
• P(Blue) = 3/10 = 30%
• P(Green) = 2/10 = 1/5 = 20%

Using Probability to Predict

Once you know the probability, you can make predictions!

Example: In our bag, 50% of draws were red. If we draw 100 times, how many reds do we expect?

• 50% of 100 = 50 red draws (expected)

Remember: It's a prediction, not a guarantee!

Why Don't They Match?

Theoretical probability (what should happen) doesn't always match experimental probability (what does happen) — and that's OK!

Example: Flip a coin 10 times

• Theoretical: Should get 5 heads
• Experimental: Might get 6 heads, or 4 heads, or even 7 heads!

The more times you try, the closer experimental gets to theoretical!

For Junior High Students

Theoretical vs. Experimental Probability

Theoretical probability: Based on mathematical analysis of all possible outcomes

• Calculated before any experiment
• Example: P(heads) = 1/2 for a fair coin

Experimental probability: Based on actual collected data

• Calculated after performing trials
• Example: Got 18 heads in 30 flips → P(heads) = 18/30 = 3/5 = 60%

Formula for experimental probability:

P(event) = frequency of event / total number of trials

Why They Differ

Theoretical probability represents the long-run behavior. In the short term, random variation causes experimental results to differ.

Example: Fair die (theoretical P(4) = 1/6 ≈ 16.7%)

Trial 1: Roll 12 times, get 1 four → experimental P(4) = 1/12 ≈ 8% Trial 2: Roll 12 times, get 3 fours → experimental P(4) = 3/12 = 25% Trial 3: Roll 120 times, get 19 fours → experimental P(4) = 19/120 ≈ 15.8%

Notice: With more trials, experimental probability gets closer to theoretical.

The Law of Large Numbers

As the number of trials increases, the experimental probability tends to get closer to the theoretical probability.

Key insight: A small number of trials can give misleading results. More data gives better estimates.

Example: Coin flips

• 10 flips: might get 7 heads (70%) — far from expected 50%
• 100 flips: might get 56 heads (56%) — closer to 50%
• 1,000 flips: might get 502 heads (50.2%) — very close to 50%

This doesn't mean experimental will ever exactly equal theoretical — just that they converge over many trials.

Frequency Tables

A frequency table organizes data by counting how many times each outcome occurs.

Components:

• Outcome/Category: The possible results
• Frequency: Count of how many times each outcome occurred
• Total: Sum of all frequencies

Example: Students' number of siblings

Raw data: 0, 1, 1, 2, 1, 3, 0, 1, 2, 1, 0, 1, 2, 1, 0, 2, 1, 1, 3, 0

Calculating Probability from Frequency Tables

Use the experimental probability formula:

P(outcome) = frequency of outcome / total frequency

From the siblings example:

P(1 sibling):

P(1) = 9/20 = 0.45 = 45%

P(0 siblings):

P(0) = 5/20 = 1/4 = 0.25 = 25%

P(2 or more siblings):

Frequency of 2 or more = 4 + 2 = 6
P(2 or more) = 6/20 = 3/10 = 30%

Relative Frequency

Relative frequency is another name for experimental probability. It's the frequency expressed as a fraction, decimal, or percentage of the total.

Note: Relative frequencies always sum to 1 (or 100%).

Making Predictions from Data

Use experimental probability to predict future outcomes.

General formula:

Expected occurrences = probability × number of trials

Example: In a survey, 30 out of 200 customers bought dessert.

Experimental probability: P(buy dessert) = 30/200 = 0.15 = 15%

Prediction: Out of the next 500 customers, how many will buy dessert?

Expected buyers = 0.15 × 500 = 75 customers

Example: A basketball player has made 48 out of 60 free throws this season.

Experimental probability: P(make free throw) = 48/60 = 0.8 = 80%

Prediction: In the next 25 attempts, how many will they make?

Expected makes = 0.8 × 25 = 20 free throws

Predictions are estimates, not certainties. Actual results will vary.

Two-Way Frequency Tables

For situations with two categories, use a two-way table (also called a contingency table).

Example: Survey of 100 students about sports and homework

Finding probabilities:

P(plays sports) = 45/100 = 45%

P(homework > 2 hrs) = 55/100 = 55%

P(plays sports AND homework > 2 hrs) = 25/100 = 25%

Conditional probability: Given a student plays sports, what's the probability they do homework > 2 hrs?

P(homework > 2 | plays sports) = 25/45 ≈ 56%

Real-Life Applications

Marketing: "20% of website visitors make a purchase"

• Predict sales based on traffic
• 10,000 visitors → expect ~2,000 purchases

Medicine: "30% of patients respond to treatment"

• Predict treatment success rates
• 200 patients → expect ~60 to respond

Quality control: "2 out of 1,000 products are defective"

• P(defective) = 2/1,000 = 0.2%
• Production of 50,000 → expect ~100 defects

Sports: Player batting average .300 (30% success)

• In 20 at-bats → expect ~6 hits

Weather: "Historical data shows rain 40% of days in April"

• Predict ~12 rainy days in a 30-day April

Accuracy of Predictions

Factors affecting accuracy:

Sample size: Larger samples give more reliable probabilities

• 10 trials: high variability
• 1,000 trials: low variability

Randomness: Events must be random and independent

• Past outcomes don't affect future ones

Conditions: Situation must remain consistent

• Weather patterns can shift
• Player performance can improve/decline

Common Mistakes

Mistake 1: Confusing theoretical and experimental probability

❌ "I flipped 10 heads in a row, so the next flip is more likely tails" ✓ Each flip is independent, still 50/50

Mistake 2: Using too few trials

❌ Roll die 6 times, no 4s appear → conclude P(4) = 0 ✓ Need many more trials to estimate accurately

Mistake 3: Not adding to 1

❌ Relative frequencies sum to 0.97 (calculation error) ✓ Relative frequencies must sum to 1.00

Mistake 4: Treating predictions as guarantees

❌ "Exactly 15% of next 500 customers will buy" ✓ "Approximately 15%, so around 75 customers"

Tips for Success

Tip 1: Clearly label frequency tables with totals

Tip 2: Check that all frequencies sum to the total

Tip 3: Use the law of large numbers — collect more data for better estimates

Tip 4: When making predictions, show your work: probability × trials

Tip 5: Remember that experimental probability is an estimate based on data

Practice