The one question to ask
When you are working with two events, ask: does knowing the first one happened change the chance of the second? If the answer is no, the events are independent. If the answer is yes, they are dependent.
That single question decides how you calculate the chance that both happen.
Independent events: coins
A fair coin has no memory. Flip it once and you get heads with probability 1/2. Flip it again and the chance of heads is still 1/2, no matter how the first flip landed. The two flips are independent.
For independent events, the chance of both happening is the product of the separate chances:
P(A and B) = P(A) × P(B)
So the chance of two heads in a row is 1/2 × 1/2 = 1/4. The chance of three heads in a row is 1/2 × 1/2 × 1/2 = 1/8. These are the chances before you start flipping. Once you have already seen two heads, the chance of heads on the third flip is back to 1/2, not 1/8. The earlier results do not change it.
Rolling dice behaves the same way. Rolling a six on one die has no effect on a second die, so the chance of two sixes is 1/6 × 1/6 = 1/36.
Dependent events: cards
Now consider a standard deck of 52 cards, from which you draw two cards without putting the first one back. The deck has changed after the first draw, so the second draw depends on the first.
Suppose you want two aces.
- On the first draw, there are 4 aces in 52 cards, so the chance is 4/52 = 1/13.
- If the first card was an ace, 3 aces remain in 51 cards, so the chance of a second ace is 3/51 = 1/17.
For dependent events, you multiply the first chance by the chance of the second given that the first happened:
P(A and B) = P(A) × P(B given A)
So the chance of two aces is 4/52 × 3/51 = 12/2652 = 1/221, which is about 0.45%.
Compare this with putting the first card back and reshuffling. Then the deck is whole again, the draws are independent, and the chance is 1/13 × 1/13 = 1/169, about 0.59%. Removing the card changes the answer, which is the whole point of the distinction.
A second card example: hearts
The same pattern applies to suits. A deck has 13 hearts.
- The chance the first card is a heart is 13/52 = 1/4.
- If it was a heart and is not replaced, 12 hearts remain in 51 cards, so the next card is a heart with probability 12/51 = 4/17.
- The chance of two hearts in a row is 13/52 × 12/51 = 156/2652 = 1/17.
Notice that 4/17 is a little lower than 1/4, because you have taken a heart out of the deck. If the first card was not a heart, the next heart would be slightly more likely than 1/4, since all 13 hearts are still in a deck that now has 51 cards. The conditional idea behind this has its own explanation under conditional probability.
Replacement is the usual switch
Many textbook examples are really about replacement:
- With replacement: the item goes back, the pool is restored, and the draws are independent.
- Without replacement: the item stays out, the pool shrinks, and the draws are dependent.
Dealing cards from a deck is nearly always without replacement, which is why card games reward careful attention to what has already been played. The effect on each draw is covered in detail under without replacement. A skilled counter in a game like blackjack makes use of this dependence, which is the starting point for card counting, although casinos take steps against it and it carries no promise of profit.
Common mix-ups
Mixing up “independent” with “mutually exclusive”. Independent events can both happen, and one does not change the other. Mutually exclusive events cannot both happen at all, so they are in fact strongly linked: if one happens, the chance of the other drops to zero.
Assuming streaks create dependence. Five reds in a row on a fair wheel or five tails on a fair coin do not make the next result more or less likely. Believing otherwise is the gamblers fallacy. If the events are independent, the earlier results are irrelevant to the next one.
Forgetting that a real source of dependence may exist. A shuffled deck has dependence, but a coin does not. Some physical systems are more complicated, such as a shoe of cards that is partly used up. Before assuming independence, think about whether something has actually been removed, changed or carried over.
A quick decision guide
- Ask whether the first outcome changes the setup for the second.
- If it does not, multiply the original chances.
- If it does, adjust the second chance to reflect what is left, then multiply.
- Check that your final number is between 0 and 1 and that it is smaller than either single chance.
A probability tree is a handy way to organise this when there are several steps, and it is explored further under probability trees. Remember that these figures describe chances before the draws, and the real outcome of any particular round remains uncertain.