Independent events, in full
Independence is a statement about information. A and B are independent if learning that A happened tells you nothing about whether B will happen, written P(B given A) = P(B). The multiplication rule follows from that: P(A and B) = P(A) x P(B). Two fair coin flips are independent, so two heads in a row has probability 0.5 x 0.5 = 0.25. The rule holds only under independence, and applying it where it does not hold is one of the most common sources of wrong probability answers.
Successive lottery draws are independent of each other. The machine carries no state from Saturday to Wednesday, the balls go back in, and last week's result has no influence on this week's. Within a single draw, the balls are not independent. They come out without replacement, so once ball 7 has been drawn it cannot appear again, and the pool shrinks by one before every subsequent pull. That dependence is exactly what the C(n,k) formula is built to account for.
The arithmetic makes the difference visible. In a 6-from-49 draw the chance the first ball is 7 is 1/49. Given that it was, the chance the second is 12 is 1/48, not 1/49, because only 48 balls remain in the drum. Multiplying in that conditional form is correct and gives 1 in 10,068,347,520 for one exact ordered sequence. Treating the six pulls as independent and using (1/49)^6 would give 1 in 13,841,287,201, which is simply wrong.
- Probability
- Statistical independence
A worked example
Two draws of the same 6-from-49 game. Hitting the jackpot in draw one has probability 1/13,983,816, and so does draw two, and because the draws are independent the chance of doing both is that number squared, roughly 1 in 195,500,000,000,000. Inside a single draw the numbers do not multiply that way, because the second ball comes from 48 remaining balls rather than 49.
See also
Where it comes up on this site
Independent events: frequently asked
Yes. Every draw starts from a full set of balls and a machine that remembers nothing.
Because they are drawn without replacement. After the first ball comes out there are 48 left instead of 49, and that ball cannot reappear, so the probability attached to every later ball depends on what has already gone. That is the definition of dependence, and it is why the count of possible tickets uses C(n,k) rather than a plain power.