Matemáticas
What Are Independent Events?
Two coin tosses, two draws with replacement, and the reason independent events and mutually exclusive events are opposites rather than the same thing.
Independent events
Two events are independent when the outcome of one has no effect on the probability of the other, so the probability of both equals the two probabilities multiplied.
- Dónde lo encuentra el alumnado
- Grade 8 probability, when tree diagrams first appear, and again in GCSE Higher and IGCSE Extended questions contrasting drawing with and without replacement.
La respuesta corta
Two events are independent when one happening does not change the probability of the other, so their probabilities multiply: P(two heads) = 1/2 × 1/2 = 1/4. Drawing counters with replacement gives independent events; drawing without replacement does not, because the second draw has fewer counters.
Un ejemplo
P(head, then head) = 1/2 × 1/2 = 1/4
The coin has no memory of the first toss, so the second toss is still an even chance and the two probabilities multiply. Confirm it by listing the four equally likely outcomes — HH, HT, TH, TT — of which one is two heads. This also settles the gambler's question: after five heads in a row, the probability of a head on the sixth toss is still 1/2, because the coin has not been keeping score.
Independence is about information, not distance
It is easy to assume that two events are independent because they happen in different places or at different times. That is not the test. The test is whether knowing the result of one changes what you would say about the other.
Two coin tosses pass, and so does the weather in Lahore compared with the roll of a die in the same room. But two events can happen simultaneously and still be independent, and two events separated by years — a student's Grade 8 marks and their Grade 10 marks — are clearly not.
The word "replacement" decides the question
A bag holds 5 red counters and 3 blue, eight in total. Take one, note the colour, put it back, and take another. The bag is identical both times, the events are independent, and P(two reds) = 5/8 × 5/8 = 25/64.
Now do it without replacing. After a red is taken, seven counters remain and only four of them are red, so the second probability is 4/7 rather than 5/8. Then P(two reds) = 5/8 × 4/7 = 20/56 = 5/14. Every question of this type turns on one word, and it is worth underlining it before starting.
On a tree diagram the difference is visible immediately: with replacement, the second set of branches repeats the first; without, the denominators drop by one.
The pair students swap round
Independent and mutually exclusive get confused constantly, and they are almost opposites. Mutually exclusive events cannot both happen; independent events have no effect on each other, which usually means they can happen together perfectly happily.
One die roll makes the contrast sharp. "Even" and "a 5" are mutually exclusive. They are certainly not independent: told the roll was a 5, the probability of even drops from 1/2 to 0. Any two mutually exclusive events with non-zero probabilities are dependent for exactly this reason, which is why one set of events is added and the other multiplied.
Preguntas frecuentes
Are independent events mutually exclusive?
No, and generally they are the reverse. Independent events can both occur — two coins can both land heads, and 1/2 × 1/2 = 1/4 is the probability that they do. Mutually exclusive events cannot both occur at all, so their joint probability is 0 rather than the product.
How do you check whether two events are independent?
Compare P(A and B) with P(A) × P(B). If they match, the events are independent; if not, they are dependent. Equivalently, check whether P(A given B) equals P(A) — whether being told that B happened changes what you would say about A.
Are draws without replacement independent?
No. Removing a counter changes both the number left in the bag and the mix of colours, so the second probability depends on what the first draw produced. Those second-stage values are conditional probabilities, and the branches of a tree diagram carry them.
After four heads in a row, is a tail more likely?
No. The coin is unchanged, so the fifth toss is still 1/2 either way. The confusion comes from a true fact used wrongly: five heads in a row has probability 1/32 before you start tossing. Once four have landed, only one toss remains uncertain and its probability is 1/2.
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