Probability Simulator
Toss coins, roll dice, spin your own spinner or draw cards — once with animation or 10,000 times in a flash — and watch experimental probability close in on theoretical probability.
Theoretical probability is P(E) = favourable outcomes ÷ total outcomes, so rolling a 5 on a die gives P = 1/6 ≈ 0.167. Experimental probability is successes ÷ trials — 27 heads in 50 tosses is 27/50 = 0.54. The more trials you run, the closer the experimental value moves to the theoretical one.
Experiment
A weight of 2 makes that sector twice as big — and twice as likely.
One card is drawn from a full 52-card deck and put back before the next draw.
Run trials
Experimental vs theoretical
P(experimental) = times it happened ÷ total trials
Results table
| Outcome | Frequency | Experimental P | Theoretical P | Difference |
|---|---|---|---|---|
| Total | 0 | — | 1 |
Law of large numbers
How the experimental probability of one outcome changes as trials pile up.
Chance you can see
Theoretical probability is what should happen; experimental probability is what did happen. Run enough trials and the two meet.
Pick an experiment
Coins, dice, a spinner you design, or a card drawn from a deck.
Predict first
Look at the theoretical column and guess what the bars will look like.
Run trials
Watch one animated trial, then run 100, 1,000 or 10,000 at once.
Compare
Check how the difference column and the convergence graph shrink as trials grow.
Theoretical probability
P(E) = favourable outcomes ÷ total outcomes
Getting a 5 on a die: 1 favourable out of 6 equally likely outcomes, so P = 1/6 ≈ 0.167. Always 0 ≤ P(E) ≤ 1, and P(not E) = 1 − P(E).
Experimental probability
P(E) = times E happened ÷ number of trials
If a coin shows heads 27 times in 50 tosses, the experimental probability of heads is 27/50 = 0.54 — close to, but not exactly, 1/2.
Two dice and 36 outcomes
Two dice give 6 × 6 = 36 equally likely pairs. A sum of 7 can happen 6 ways — (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) — so P(7) = 6/36 = 1/6, the most likely sum. A sum of 2 or 12 has only 1/36.
A deck of 52 cards
4 suits × 13 ranks. Hearts ♥ and diamonds ♦ are red; spades ♠ and clubs ♣ are black. There are 12 face cards (J, Q, K of each suit) and 4 aces, so P(face card) = 12/52 = 3/13 and P(ace) = 1/13.
Probability Simulator — your questions
What is the formula for probability?
P(E) = number of favourable outcomes ÷ total number of equally likely outcomes. Every probability lies between 0 and 1, where 0 means impossible and 1 means certain. The probability of an event not happening is P(not E) = 1 − P(E), so if P(rain) = 0.3 then P(no rain) = 0.7.
What is the difference between experimental and theoretical probability?
Theoretical probability is worked out by counting outcomes, so a fair coin gives P(head) = 1/2. Experimental probability comes from actually doing the experiment: 27 heads in 50 tosses gives 0.54. They rarely match exactly for a small number of trials, but they get closer as the trials increase.
What is the probability of getting a sum of 7 with two dice?
Two dice give 6 × 6 = 36 equally likely pairs. A sum of 7 happens six ways — (1,6), (2,5), (3,4), (4,3), (5,2) and (6,1) — so P(7) = 6/36 = 1/6. That makes 7 the most likely sum, while 2 and 12 each have only 1/36.
What is the probability of drawing a face card from a deck?
A standard deck has 52 cards in 4 suits of 13. The face cards are the jack, queen and king of each suit, so there are 12 of them and P(face card) = 12/52 = 3/13. There are 4 aces, so P(ace) = 1/13, and 26 red cards, so P(red) = 1/2.
Why did I get 7 heads out of 10 tosses?
That is completely normal. With only 10 tosses, results far from 5 heads happen often just by chance. The law of large numbers says the experimental probability settles near the theoretical value only when the number of trials is large — try the same coin with 10,000 tosses and watch it close in on 0.5.
Can I use this simulator for a school project?
Yes. Run a set number of trials, note the frequency table and the experimental probability for each outcome, then compare them with the theoretical values and write about the difference. The convergence graph makes a good illustration of why larger samples give more reliable results.