Combined-Conditional-Probabilities Aqa Higher

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Combined conditional probabilities

What is a combined conditional probability?

This is when you have two (or more) successive events, one after the other, and the second event depends on (is conditional on) the first

How do I calculate combined conditional probabilities?

You need to adjust the number of outcomes as you go along
- For example, selecting two cards from a pack of 52 playing cards without replacing the first card:
  - P(red 1^st card) is 26 reds out of 52 cards
  - If the 1^st card is not replaced, there are only 25 reds left out the remaining 51 cards
  - P(red 2^ndcard) is 25 reds out of 51 cards
  - P(red then red) = $26 over 52 cross times 25 over 51$

Examiner Tips and Tricks

If a question says “two cards are drawn” then you may assume that they draw 1 card followed by another card without replacement (the maths is the same).

Can I draw a tree diagram for combined conditional probabilities?

Yes, a tree diagram is a useful way to show combined conditional probabilities
- For example, two counters are drawn at random from a bag of 3 blue and 8 red counters without replacement
  - The probabilities are shown below

What if there are multiple possibilities within one question?

You may need a listing strategy (e.g. AAB, ABA, BAA)
You will need the or rule for multiple possibilities
- P(AB or BA or AA or…) = P(AB) + P(BA) + P(AA) +…
  - Add the cases together
Remember that AB and BA are not the same
- AB means A happened first, then B
- BA means B happened first, then A

Examiner Tips and Tricks

Try not to simplify your probabilities too early as it is easier to add probabilities together when they all have the same denominator!

Worked Example

A bag contains 10 yellow beads, 6 blue beads and 4 green beads.

A bead is taken at random from the bag and not replaced.

A second bead is then taken at random from the bag.

(a) Find the probability that both beads are different colours.

Let Y, B and G represent choosing a yellow, blue and green bead

Method 1

The probability of the beads being different colours is equal to 1 subtract the probability that the beads are the same colour

Find the probability of both beads being the same colour

P(same colours) = P(YY) + P(BB) + P(GG)

Calculate each conditional probability separately, remembering the number of beads changes after one is drawn and not replaced

For example, P(YY) = $10 over 20 cross times 9 over 19$

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