Probability Guide: Independent Events and the Binomial

Two events are independent when one happening does not change the probability of the other, and dependent when it does. That single distinction decides which formula you use: independent events multiply, dependent events need a conditional probability, and events that cannot both happen (mutually exclusive) simply add. Probability is always a number from 0 (impossible) to 1 (certain), often written as a percentage.

Which rule applies to which situation?

The probability calculator on this site covers the independent and mutually exclusive cases for two events. For P(A) = 0.3 and P(B) = 0.2 as independent events it gives P(A and B) = 0.06 and P(A or B) = 0.44, matching the table. It does not model dependent events, so those are worked by hand below.

How does dependence change the answer? Drawing cards

Draw two cards from a standard 52-card deck and ask for two aces. If you put the first card back, the draws are independent. If you keep it, the second draw depends on the first.

Removing the first ace leaves 3 aces in 51 cards, which is why the second draw has a lower chance than the 4/52 (7.69%) it started with. The same logic applies whenever you sample from a small finite pool without putting items back, such as inspecting items from a small batch.

How do you count outcomes with combinations?

When order does not matter, the number of ways to choose k items from n is n! / (k! x (n - k)!). Choosing 3 people from 10 gives 120 groups. A 5-card hand from a 52-card deck can be chosen in 2,598,960 ways. A game that draws 6 numbers from 49 has 13,983,816 equally likely combinations, so any one specific ticket has a 1 in 13,983,816 chance (about 0.0000072%), assuming a fair draw.

What is the binomial distribution?

Use it when you repeat the same independent trial n times, each with success probability p, and count the successes. The chance of exactly k successes is C(n, k) x p^k x (1 - p)^(n - k).

Note the third row: 'at least one' is almost always easiest as one minus the chance of none. Six rolls do not guarantee a six, and they give about two chances in three, not certainty.

What is conditional probability, in a worked example?

Suppose, as a made-up quality-control scenario, that 1% of items are defective, a test flags 95% of defective items, and it wrongly flags 5% of good items. Take 10,000 items: 100 are defective and the test flags 95 of them. Of the 9,900 good items it wrongly flags 495. That is 590 flagged items in total, so the probability that a flagged item is actually defective is 95 / 590 = 16.1%.

Most people expect something near 95%, because the test sounds accurate. The low answer comes from the base rate: defects are rare, so false alarms from the large good group outnumber real hits. This is Bayes' theorem applied with counts, and the same reasoning applies to any screening or filtering system whose target is uncommon. The numbers here are illustrative and are not a statement about any real test.

Common mistakes

• Gambler's fallacy: after 5 heads in a row, the next fair flip is still 50%. Five heads in a row had a 3.125% chance beforehand, but past flips do not change the next one.
• Confusing mutually exclusive with independent. Two events with non-zero probabilities that cannot both happen are dependent, because knowing one happened tells you the other did not.
• Adding probabilities for 'or' without subtracting the overlap, which can push the total above 100%.
• Assuming independence by default. Weather on consecutive days, or components sharing one power supply, are rarely independent.
• Reading a percentage as a guarantee. A 66.5% chance still fails one time in three.