Theoretical probability is worked out by counting outcomes. A fair coin gives a half.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Experimental probability
Flip it 10 times and get 7 heads, and the experimental probability is 7/10.
A gap is expected
7 heads in 10 does not make the coin unfair. Small samples wander.
More trials tighten it
Over a thousand flips the fraction settles close to a half. That is what "long run" means.
Two kinds of probability
Theoretical probability comes from reasoning about the situation: a fair die gives 1/6. Experimental probability comes from actually doing it and counting: 23 sixes in 120 rolls gives about 0.19. They answer the same question by different routes.
More trials, closer agreement
With 10 rolls the experimental value can be far from 1/6. With 6,000 it will be close. This is the law of large numbers: as trials increase, the experimental probability settles towards the theoretical one.
When they disagree persistently
If thousands of trials keep giving results far from the theory, the theory is probably wrong — the die may be loaded, or the outcomes may not be equally likely. Persistent disagreement is evidence about the world, not a failure of the experiment.
Why both are needed
Some situations have no calculable theory — the chance a particular bridge design fails, or that a new medicine works. There you must experiment. Theoretical probability is a luxury available only when the structure is known.
Step 2: Try It Yourself
Set the counts you observed and compare them.
Heads
Tails
There are 10 in all.
Heads has the most. It has 4 more than Tails.
Step 3: Watch an Example
One step at a time.
Watch Leo Judge a Coin
Leo flips a coin 10 times and gets 7 heads.
- Step 1
The theoretical probability of heads is 1/2, so he expected about 5.
Step 4: Your Turn
Practice makes it stick.
The Flips
Problem 1 of 2
20 flips give 12 heads. How many gave tails?
The Expectation
Problem 2 of 2
A fair coin flipped 60 times. About how many heads are expected?
Expected and Observed
1 of 4
A fair die rolled 60 times. About how many sixes are expected?
2 of 4
50 spins give 30 reds. How many were not red?
3 of 4
Does 7 heads in 10 prove a coin is unfair? 1 for yes, 0 for no.
4 of 4
Do more trials usually bring the two closer? 1 for yes, 0 for no.
Step 5: Quick Check
Show what you know.
Question 1 of 1
A fair die rolled 120 times. About how many threes are expected?
What You Learned
- Theoretical probability is counted; experimental probability is observed.
- A gap between them is normal, especially with few trials.
- The more trials, the closer the observed fraction usually gets.