E8.3 Probability of combined events

Learning objectives

E8.3.1Calculate the probability of combined events using, where appropriate:• Calculate the probability of combined events using, where appropriate: • sample space diagrams • Venn diagrams • tree diagrams. Combined events could be with or without replacement. The notation P(A ∩ B) and P(A ∪ B) may be used in the context of Venn diagrams. On tree diagrams outcomes will be written at the end of branches and probabilities by the side of the branches.

A combined event joins conditions or stages. Choose a representation that lists every possible outcome once, then combine only the outcomes that satisfy the event.

Representation	Best use	Probability method
sample space diagram	two stages fit rows and columns	mark favourable cells; count cells only when they are equally likely
two-set Venn diagram	outcomes belong to $A$ , $B$ , both or neither	$P(A\cap B)$ uses the overlap; $P(A\cup B)$ uses every region in $A$ or $B$ , counting the overlap once
tree diagram	outcomes occur in stages	multiply along each required path; add probabilities of alternative paths

P(\text{one path})=\text{product of its branch probabilities},\qquad P(\text{alternative paths})=\text{sum of their path probabilities}

Selection	What happens before the next branch?
with replacement	the selected item returns, so totals and probabilities reset
without replacement	the selected item stays out, so the total falls by 1 and the relevant category may also fall by 1

A bag has 5 green and 3 blue counters. With replacement, $P(\text{two green})=\tfrac58\times\tfrac58=\tfrac{25}{64}$ . Without replacement, after one green there are 4 green among 7 counters, so $P(\text{two green})=\tfrac58\times\tfrac47=\tfrac5{14}$ .

On a tree diagram, write each outcome at the end of its branch and its probability beside the branch. Check that probabilities leaving the same point add to 1. In a Venn diagram, 'and' means the intersection $\cap$ and inclusive 'or' means the union $\cup$ ; the overlap must not be counted twice.

Do not use a fixed second-stage denominator in a without-replacement problem. Conditional probability notation $P(A\mid B)$ and its formulas belong to E8.4, not this card.

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—E8.4 Conditional probabilityRead