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Approximate Probability
From Collected Data

Learn how to collect data from experiments and use relative frequency to approximate the probability of real-world events.

7th Grade Math

What Is Probability?

Probability measures how likely an event is to happen. It is always a number between \(0\) and \(1\):

\(P = 0\)
Impossible: the event will never happen.
\(P = 0.5\)
Equally likely: a coin flip landing heads.
\(P = 1\)
Certain: the event will always happen.

Theoretical vs Experimental Probability

Theoretical
What should happen based on math. You calculate it without doing any experiment.
Experimental
What actually happens when you collect data. You find it by running trials.

What Is Relative Frequency?

Relative frequency is the fraction of times an event occurs out of the total number of trials. It is our way of approximating probability from data.

Relative Frequency =
\(\frac{\text{Number of times event occurs}}{\text{Total number of trials}}\)

The Experimental Probability Formula

This formula lets you approximate the probability of any event just by collecting data and counting. The symbol \(\approx\) means "approximately equal to."

1
Run the experiment many times (these are your trials).
2
Count how many times the event you care about happened.
3
Divide that count by the total number of trials.

The Law of Large Numbers

As you repeat an experiment more and more times, the experimental probability gets closer and closer to the true (theoretical) probability.

Relative Frequency
Number of Trials
 0 0.25 0.50 0.75 10 50 200 600
\(P = \frac{1}{6}\)
 Wild swings Settles near P

Exact vs Approximate

Theoretical Probability Experimental Probability
Source Mathematical reasoning Collected data / experiments
Precision Exact value Approximation (uses \(\approx\))
Example \(P(\text{heads}) = \frac{1}{2}\) Flipped 100 times, got 47 heads: \(\frac{47}{100}\)
Requires Equally likely outcomes Only data collection

Predicting From Probability

If you know (or have approximated) a probability, you can predict how many times an event should occur in a given number of trials:

Example: A number cube has \(P(3 \text{ or } 6) = \frac{2}{6} = \frac{1}{3}\). In 600 rolls, you predict about \(\frac{1}{3} \times 600 = \) 200 rolls of 3 or 6.

Step-by-Step Approach

1
Identify the event you want to find the probability of.
2
Collect data: perform the experiment many times and record results.
3
Count favorable outcomes: how many times did the event happen?
4
Calculate: divide favorable outcomes by total trials to get the relative frequency.
5
Predict: multiply the probability by any new number of trials to predict future results.

Example 1: Coin Flip Experiment

Maya flipped a coin 80 times and got heads 36 times. What is the experimental probability of heads?

Heads Observed 36 favorable outcomes Total Flips 80 total trials

Example 2: Spinner Experiment

A spinner has 4 equal sections: Red, Blue, Green, and Yellow. Leo spun it 200 times:

Color Observed Relative Frequency
Red 55 \(\frac{55}{200} = 0.275\)
Blue 48 \(\frac{48}{200} = 0.24\)
Green 52 \(\frac{52}{200} = 0.26\)
Yellow 45 \(\frac{45}{200} = 0.225\)

Example 3: Predicting Outcomes

A number cube is rolled 600 times. Predict how many times a 3 or 6 will appear.

1
Find the probability: A number cube has 6 faces. Two faces show 3 or 6, so \(P(3 \text{ or } 6) = \frac{2}{6} = \frac{1}{3}\).
2
Multiply: \(\frac{1}{3} \times 600 = 200\).
3
Interpret: You predict about 200 rolls, but probably not exactly 200.

Example 4: Free-Throw Data

Jaylen made 42 out of 60 free throws this season. Predict how many he will make in his next 25 attempts.

1
Experimental probability: \(P(\text{make}) \approx \frac{42}{60} = 0.7\).
2
Predict: \(0.7 \times 25 = 17.5\).
3
Interpret: Jaylen should make about 17 or 18 free throws. We round because you cannot make half a shot.

Why More Trials Matter

10 Trials
Heads: \(\frac{7}{10}\)
 70% Far from 50% 100 Trials
Heads: \(\frac{53}{100}\)
 53% Getting closer 1,000 Trials
Heads: \(\frac{498}{1000}\)
 49.8% Very close to 50% → →

Common Mistakes

✕ Expecting Exact Results
Thinking 600 rolls must give exactly 200 threes or sixes. Probability predicts "about," not "exactly."
✕ Too Few Trials
Drawing conclusions from only 5 or 10 trials. Small samples produce unreliable approximations.
✕ Flipping the Fraction
Writing \(\frac{\text{total trials}}{\text{event count}}\) instead of \(\frac{\text{event count}}{\text{total trials}}\). The event goes on top.

Summary Table

Concept Key Formula / Idea Remember
Experimental Probability \(\frac{\text{favorable outcomes}}{\text{total trials}}\) Based on real data
Theoretical Probability \(\frac{\text{favorable outcomes}}{\text{possible outcomes}}\) Based on math/logic
Prediction \(P \times \text{trials}\) Says "about," not "exactly"
Long-Run Frequency More trials → closer to true \(P\) Law of Large Numbers

Question 1 of 5

A bag contains colored marbles. Sam drew a marble 50 times (replacing each time) and got red 18 times. What is the experimental probability of drawing red?

Question 2 of 5

A number cube is rolled 300 times. About how many times would you predict rolling a number less than 3 (i.e., 1 or 2)?

Question 3 of 5

Rosa flipped a coin 10 times and got tails 8 times. She says the probability of tails is \(0.80\). What is wrong with her conclusion?

Question 4 of 5

A basketball player has an experimental free-throw probability of \(0.6\). If she shoots 40 free throws in practice, about how many will she make?

Question 5 of 5

Which experiment would give the most reliable experimental probability for the chance of rain on a given day in April?