Probability Calculator

How do you calculate probability?

A single probability is the count of favourable outcomes divided by the count of all possible outcomes. Take one die as an example: one face out of six gives the fraction 1/6, a decimal of 0.166667, 16.67% and exact odds of 1:5. The complement, meaning every roll that is not a six, takes the remaining 83.33%.

That division holds only while every outcome carries the same weight, which covers a fair die or a shuffled deck. Drawing an ace from a standard 52-card deck counts 4 favourable cards out of 52, reduced to 1/13, worth 7.69% at odds of 1:12. A loaded die breaks the rule. Once one face turns up 30% of the time, counting faces gives the wrong answer and the measured 0.3 has to go straight into a probability field.

Are two events independent or mutually exclusive?

The two-event mode assumes independence throughout, meaning one event happening leaves the odds of the other untouched. It multiplies for the joint case, written P(A and B) = P(A) × P(B). For the union it adds the two probabilities then takes the overlap back out, written P(A or B) = P(A) + P(B) − P(A) × P(B).

With A at 0.5 and B at 0.25 the results table returns 12.5% for both happening, 62.5% for either one, 37.5% for neither and 50% for exactly one of the two. At least one lands on 62.5% as well. It is the same question from the other side, 1 minus the 37.5% chance that neither shows up.

Mutually exclusive events cannot both land on the same trial, which rewrites both formulas. Their joint probability is 0 and their union becomes a plain sum, P(A or B) = P(A) + P(B). One card that is an ace or a king is the textbook case, worth 8 out of 52 at 15.38%.

Feed those two 7.69% events into the two-event mode and the answer comes back at 14.79%, short by 0.59 points, which is exactly the phantom overlap it subtracted. Staying in the single-event mode and entering 8 favourable out of 52 returns the true 15.38%, at the fraction 2/13 and exact odds of 2:11.

What does the binomial probability calculator work out?

Ten flips of a coin can land three heads in C(10, 3) = 120 different orders, which the binomial mode turns into an exact probability of 11.72%, a cumulative 17.19% for at most three heads and 94.53% for at least three. The mean is 5 heads with a standard deviation of 1.58, so three heads falls 1.3 deviations below the middle of the binomial distribution.

A player shooting 80% from the free-throw line hits exactly 4 of 5 attempts on 40.96% of trips, while 4 or more comes up 73.73% of the time. Any run of repeated attempts at a fixed probability with two outcomes each fits the same mode, including a sampling check on a production batch. The calculator accepts up to 1000 trials, beyond which the binomial coefficients outgrow floating-point precision.

How do you calculate theoretical and experimental probability?

A fair coin has a 50% chance of heads on any single flip, settled by counting two faces without tossing anything. That counting on paper is theoretical probability, the job all three modes do. Experimental probability, also called empirical probability, comes from what somebody observed, dividing recorded successes by recorded trials.

Both use the same division, though the two numbers answer different questions. Flip a coin 100 times and log 47 heads and the experimental probability is 47%, which the single-event mode returns from 47 favourable out of 100. That gap against the theoretical 50% narrows as the trial count grows, an effect known as the law of large numbers.

Why show a probability four different ways?

Each row of the results table shows a decimal, a percentage, odds and a 1 in N reading, since each one suits a different setting. Statistics work runs on the decimal, a report reads better in percent while bookmakers and medical risk sheets quote odds. The 1 in N form is the one most people picture, so 0.1171875 becomes 1 chance in 8.53. Every figure prints to six decimals, so the 16.67% above reads 16.666667 on screen.

Odds of 1:12 on that ace count 4 favourable cards against the other 48, reduced to one against twelve. The probability is 1/13, while 1/12 would be 8.33%. Taking the second number of an odds pair for the total is the usual slip.