Two events are independent if the likelihood of one occurring does not depend on the other having happened.

Examples of independent events:

- flipping a coin - whether you get a head this time is not affected by what happened last time
- throwing a die - again, getting a 3, say, is not dependent on what happened last time
- winning the lottery - whether you win a prize with this week's ticket is not affected by last week's ticket
- which card I pick out of a pack of cards if I make sure that I have replaced a card I picked out earlier

Two events are dependent if the likelihood of one occurring is affected by the other.

Examples of dependent events:

- today's weather is likely to be dependent on yesterday's weather
- what time I get up in the morning is likely to depend on what time I went to bed the previous night
- which card I pick out of a pack of cards if I have already removed one card

If a coin has given a run of heads, many people believe that a tail is 'overdue'; this is known as the gambler's fallacy.

A coin has no memory, and the chance of getting a head does not change. Any single sequence of heads and tails is as likely as any other.

However, if you flip a coin three times, these are the possible outcomes:

HHH, HHT, HTH, THH, HTT, THT, TTH, TTT

Out of these possible outcomes, there are three which have two heads and one tail, and three which have two tails and one head, whereas there is only one outcome of three heads and one of three tails. This gives the **appearance** that some sequences are more likely than others, particularly where the three coins are flipped simultaneously rather than sequentially.

Another example of this misconception is that some sequences of numbers are more likely to be winning combinations in the lottery. Again, any one sequence is as likely - or unlikely - as any other. It makes no difference if you choose a sequence like 1, 2, 3, 4, 5, 6 or 17, 23, 45, 9, 11, 27 - both are equally likely (although you will have to share any prize you win with more people if you choose a popular sequence).

Another example of this misconception is that some sequences of numbers are more likely to be winning combinations in the lottery. Again, any one sequence is as likely - or unlikely - as any other. It makes no difference if you choose a sequence like 1, 2, 3, 4, 5, 6 or 17, 23, 45, 9, 11, 27 - both are equally likely (although you will have to share any prize you win with more people if you choose a popular sequence).

Mathematically, two events are independent if the probability that they both occur is equal to the product of their individual probabilities. So:

P(A and B) = P(A ∩ B) = P(A) x P(B) = P(B and A) = P(B ∩ A)

P(A and B) is what we get when we multiply the probabilities along a set of branches on a probability tree, or calculate the probability of the intersection of A and B on a Venn Diagram. Because multiplication is commutative, it does not matter whether A precedes B or B precedes A when calculating the joint probability of independent events.

If this condition does not hold, then the events are not independent.

It is also true that if A and B are independent events, then:

If this condition does not hold, then the events are not independent.

It is also true that if A and B are independent events, then:

P(A|B) = P(A) and P(B|A) = P(B)

since neither A nor B gives us any information about the other.

Sampling with replacement may mean that the events are independent, although this is not necessarily the case. For Prize Giving, Ingrid's method involves sampling with replacement, and here the events of choosing a 1st prize winner and a 2nd prize winner are independent of each other.

Sampling without replacement generally means that the events are not independent. In the case of Prize Giving, Paolo's method involves sampling without replacement. If we look at the probability of choosing two girls:

P(G and G) = $\frac{6}{8}\times\frac{5}{7}=\frac{15}{28}$

However,

P(G) x P(G) = $\frac{6}{8}\times\frac{6}{8}=\frac{9}{16}>\frac{15}{28}$

The resource Prize Giving provides a context for discussion of independence and dependence, and an opportunity for students to explore how different sampling methods affect the chance of an individual being chosen, or of a 'representative' sample being chosen - which also raises the question of what we mean by 'representative'.

Louis' Icecream Business is a longer investigation which also provides a context for discussing independent and dependent events - particularly the weather sequence in the final part of the problem.

Of course, in a large enough population, sampling with and without replacement will not be distinguishable, and so it may be reasonable to treat dependent events as independent in these circumstances.