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Maths

Probability - Higher

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If you are studying the higher paper, you will also need to know about tree diagrams, the and/or rule and conditional probabilities.

# Tree diagrams

Tree diagrams allow us to see all the possible outcomes of an event and calculate their probability. Each branch in a tree diagram represents a possible outcome.

If two events are independent, the outcome of one has no effect on the outcome of the other. For example, if we toss two coins, getting heads with the first coin will not affect the probability of getting heads with the second.

A tree diagram which represent a coin being tossed three times looks like this :

From the tree diagram, we can see that there are eight possible outcomes. To find out the probability of a particular outcome, we need to look at all the available paths (set of branches).

The sum of the probabilities for any set of branches is always 1.

Also note that in a tree diagram to find a probability of an outcome we multiply along the branches and add vertically.

The probability of three heads is:

P (H H H) = 1/2 × 1/2 × 1/2 = 1/8

P (2 Heads and a Tail) = P (H H T) + P (H T H) + P (T H H)

= 1/2 × 1/2 × 1/2 + 1/2 × 1/2 × 1/2 + 1/2 × 1/2 × 1/2

= 1/8 + 1/8 + 1/8

= 3/8

Question
• Bag A contains three red marbles and four blue marbles.
• Bag B contains five red marbles and three blue marbles.
• A marble is taken from each bag in turn.

Find the missing probabilities for the tree diagram:

Question

What is the probability of getting a blue bead followed by a red?

Multiply the probabilities together:

• P (blue and red) = 4/7 × 5/8
• = 20/56
• = 5/14
Question

What is the probability of getting a bead of each colour?

P (blue and red or red and blue) = P (blue and red) + P (red and blue)

• = 4/7 × 5/8 + 3/7 × 3/8
• = 20/56 + 9/56
• = 29/56

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