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Unit D3 Section 3
Tree Diagrams and Conditional Probability

The probabilities of certain events may change as a result of earlier events. For example, if it rains today, the probability that it rains tomorrow may be greater than if it was sunny today. When using a tree diagram, check that the probabilities do not change as a result of the first event. When probabilities depend on previous events they are called conditional probabilities.

Worked Examples

1

A bag contains 7 red balls and 3 green balls. Two balls are taken from the bag. Find the probabilities that they are:

(a)

both red

(b)

both green

(c)

one red and one green.

For the first ball taken from the bag,

p(Red) = and p(Green) =

If the first ball was red the bag now contains:

6 red balls and 3 green balls

so

p(Red) = and p(Green) =
= =

If the first ball was green the bag now contains:

7 red balls and 2 green balls

so

p(Red) = and p(Green) =

These probabilities can be used to create a tree diagram.

So

p(both red) =
p(both green) =
p(one red and one green) = +
=
= .
2

Weather experts estimate the probability of rain any day as

0.6 if it rained the previous day

0.3 if it did not rain the previous day.

Find the probability that a dry day is followed by,

(a)

two more dry days,

(b)

two wet days,

(c)

a wet day and then a dry day.

The tree diagram has the probabilities included on its branches. Note how they depend on the weather on the previous day.

p(two more dry days) = 0.49

p(two wet days after the dry day) = 0.18

p(dry followed by a wet day and then a dry day) = 0.12

Exercises

A bag contains 3 red counters and 4 blue counters. Two counters are taken out of the bag. Find the probability that:

(a)
both counters are red,
(b)
both counters are the same colour,
(c)
the counters are different colours.

A card is drawn at random from a full pack of playing cards. It is not put back. A second card is then drawn from the same pack.

(a)

Find the probability that:

(i)
both cards are hearts,
(ii)
only one card is a heart,
(iii)
neither card is a heart.
(b)

Find the probability that:

(i)
both cards are sevens,
(ii)
neither card is a seven,
(iii)
at least one seven is obtained.

The probability that a football team win their next match is:

⎧
⎨
⎩

0.3 if they won their last match,

0.8 if they lost their last match.

(a)

Find the probability that if a match is lost, the next two will be won.

(b)

Find the probability that if a match is won, the next two will also be won.

A pack of 24 cards contains 12 red cars, 8 yellow cards and 4 pink cards. In a game a player has to select two cards, one after the another. Find the probabilities that a player selects,

(a)
2 red cards,
(b)
2 yellow cards,
(c)
2 pink cards,
(d)
2 cards that are the same colour,
(e)
a pink card and a yellow card in any order.

Nisha buys a bag of sweets. The bag contains 3 orange, 17 strawberry and 10 cherry flavour sweets. Nisha eats 3 sweets. Find the probability that she has:

(a)
not eaten an orange sweet,
(b)
eaten 3 sweets of the same flavour,
(c)
only eaten strawberry flavour sweets.

Two yachts A and B regularly race against each other. In rough weather the probability that A wins is 0.8 and in fine weather 0.4. The probability of fine weather at a race is 0.7.

(a)

What is the probability that A wins a race?

(b)

If there is a best of 3 races competition, what is the probability that A wins the competition?

Round up to 2 decimal places.

Brian enjoys swimming. If it is a sunny day the probability that he swims is 0.9. It the day is not sunny, the probability that he swims is 0.65.

The probability that tomorrow will be sunny is 0.8.

Draw a tree diagram to illustrate this information.

Calculate the probability that Brian will not swim tomorrow.

Wes gives Bronwen a box of 25 mixed sweets. 12 of the sweets have soft centres, 8 have toffee centres and 5 have hard centres. All of the sweets have identical wrappers. Bronwen chooses 2 sweets at random.

(a)

What is the probability that Bronwen will choose 2 toffee centred sweets?

After they have eaten 3 soft centred sweets and 2 toffee centred sweets, Wes chooses 3 sweets at random from the 20 sweets that are left.
(b)

What is the probability that Wes will choose a soft centred sweet, followed by a toffee centred sweet, followed by a hard centred sweet?

When there are only 10 sweets left, 3 have soft centres, 4 have toffee centres and 3 have hard centres. Bronwen chooses two sweets at random.
(c)

What is the probability that neither of the sweets will have a hard centre?

The fast food chain Macduff's is running a competition.

You obtain a card with 12 circles covered up.

You are allowed to scratch off up to three circles.

You win if 2 or more PALM TREES are revealed.

You lose if two or more CRABS are found.

If you win, you scratch off one of the three squares to show your prize.

One of the cards is illustrated above with all the circles and squares revealed. The ratio of PALM TREES to CRABS is always 2:1 but their positions can change.

(a)

The first circle you scratch off reveals a PALM TREE. What is the probability that the second circle reveals a PALM TREE?

(b)

Copy and complete the tree diagram for all the winning combinations.

(c)

List the possible winning combinations.

(d)

What is the probability of winning?