Download PROBABILITIES FROM TREE DIAGRAMS

TREE DIAGRAMS AND BINOMIAL PROBABILITIES (Chapter 20)
Example 2
405
Self Tutor
John plays Peter at tennis. The first to win two sets wins the match.
Illustrate the sample space using a tree diagram.
If J means “John wins the set”
and P means “Peter wins the set”
then the tree diagram is:
1st set
2nd set
3rd set
J
J
J
P
P
J
We could write the sample space
in set notation as
S = fJJ, JPJ, JPP, PJJ, PJP, PPg.
J
P
P
P
2 Use a tree diagram to illustrate the sample space for the following:
a The genders of a 4-child family.
b Bag A contains red and white marbles and bag B contains blue and yellow marbles.
A bag is selected and one marble is taken from it.
c Hats A, B and C each contain pink and purple tickets. A hat is selected and then
two tickets are taken from it.
d Two teams, X and Y, play football. The first team to kick 3 goals wins the match.
e Jody and Petria play tennis. The first to win two sets in a row or a total of 3 sets,
wins the match.
3 A bag contains five marbles. One is blue, one is red, and three are green. Two marbles
are selected from the bag without replacement. Use a tree diagram to show all possible
outcomes.
B PROBABILITIES FROM TREE DIAGRAMS
Tree diagrams can be used to illustrate sample spaces providing there are not too many
different alternatives. Once the sample space has been illustrated, the tree diagram can be
used to determine probabilities.
INDEPENDENT EVENTS
of passing Physics and
We assume that these events are
independent, which means the
probability of Charles passing
Physics is not related in any way to
him passing Chemistry.
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A tree diagram showing all possible
outcomes is shown alongside with P
being a pass and F being a fail.
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Suppose Charles has probability
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of passing Chemistry.
Chemistry
outcomes probabilities
Jq_p_
P
PP
Rt_ ! Jq_p_ " Wt_Ip_
Dq_p_
F
PF
Rt_ ! Dq_p_ " Qt_Wp_
Jq_p_
P
FP
Qt_ ! Jq_p_ " Jt_p_
Dq_p_
F
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Qt_ ! Dq_p_ " Dt_p_
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TREE DIAGRAMS AND BINOMIAL PROBABILITIES (Chapter 20)
The probabilities for passing and failing each subject are marked along the branches.
When using tree diagrams to solve probability questions:
² the probability for each branch is calculated by multiplying the probabilities along that
branch
² if two or more branches meet the description of the compound event, the probability
of each branch is found and then they are added.
Example 3
Self Tutor
Jessica has probability 45 of getting an A in Mathematics and probability
an A for English.
a Display this information on a tree diagram.
b What is the probability Jessica gets one A in the two subjects?
2
5
of getting
a A is the event of getting an A. N is the event of not getting an A.
English
outcomes
Wt_
A
AA
Et_
N
AN
Rt_ ! Et_ " Qw_Wt_
Wt_
A
NA
Qt_ ! Wt_ " Sw_t_
Et_
N
NN
Mathematics
Rt_
A
Qt_
N
probabilities
b Jessica can get one A either by getting an A in Mathematics but not in English, or
by getting an A in English but not in Mathematics.
So, the total probability is 45 £ 35 + 15 £ 25 = 14
25
These events are highlighted on the tree diagram.
EXERCISE 20B.1
1 A spinner has probability 13 of finishing on blue and probability 23 of finishing on green.
Xin spins the spinner two times.
a Display this information on a tree diagram.
b Find the probability that the spinner finishes on blue once and on green once.
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2 Andrew has probability 15 of winning the 100 metre sprint and probability
the 200 metre race.
a Draw a tree diagram showing all of Andrew’s chances.
b What is the probability Andrew wins both races?
c What is the probability that Andrew wins exactly
one of the two races?
d What is the probability Andrew loses both races?
e Find the sum of the probabilities in b, c and d.
Explain why this answer is to be expected.
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TREE DIAGRAMS AND BINOMIAL PROBABILITIES (Chapter 20)
DEPENDENT EVENTS
Suppose a hat contains 5 red and 3 blue tickets. One ticket is randomly
chosen, its colour is noted, and it is then thrown away. A second ticket
is randomly selected. What is the chance that it is red?
P(second is red) =
4
7
4 reds remaining
7 to choose from
If the first ticket was blue, P(second is red) =
5
7
5 reds remaining
7 to choose from
If the first ticket was red,
R
B
B
R
R
R
B
R
So, the event of the second ticket being red depends on what colour the first ticket was.
In such a case we say we have dependent events.
Two or more events are dependent if they are not independent.
Dependent events are events where the occurrence of one of the events
does affect the occurrence of the other event.
If A and B are dependent events then
P(A then B) = P(A) £ P(B given that A has occurred).
A typical example of dependent events is when we sample two objects without replacement.
This means that the first object is not replaced before the second is selected. It therefore
cannot be selected twice.
Example 4
Self Tutor
A box contains 4 red and 2 yellow tickets. Two tickets are randomly selected one after
the other from the box, without replacement.
a Display this information on a tree diagram.
b What is the probability that both are red?
c What is the probability that one is red and the other is yellow?
a Let R be the event that a
red ticket is drawn and Y
be the event that a yellow
ticket is drawn.
Note that the outcome of
the second event depends
on the first.
1st selection
Ry_
R
Wy_
Y
2nd selection
Et_
R
Ry_ ! Et_
Wt_
Y
Ry_ ! Wt_
Rt_
R
Wy_ ! Rt_
Qt_
Y
Wy_ ! Qt_
b P(R and then R) = P(R) £ P(R given that R has occurred)
=
4
6
£
3
5
=
2
5
P(one is R and the other is Y )
= P(R and then Y or Y and then R)
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TREE DIAGRAMS AND BINOMIAL PROBABILITIES (Chapter 20)
EXERCISE 20B.2
1 A box contains 7 red and 3 green balls. Two balls are randomly selected from the box
one after the other. The first is not replaced in the box before the second is selected.
Determine the probability that:
a both are red
b the first is green and the second is red
c a green and a red are obtained.
Selecting items
simultaneously means
there is no replacement.
2 A hat contains the names of the 7 players in a
tennis squad including the captain and the vice
captain. A team of 3 players is chosen at random
by drawing the names from the hat.
a Display this information on a tree diagram.
Distinguish between the captain, the vice
captain, and the other players.
b Find the probability that the team:
i does not contain the captain
ii contains neither the captain nor the vice captain
iii contains either the captain or the vice captain, but not both.
3 Amelie has a bag containing two different varieties of apples. They are approximately
the same size and shape, but one variety is red and the other is green. There are 4 red
apples and 6 green ones. She selects one apple at random, eats it, and then takes another,
also at random. Determine the probability that:
a both apples were red
b both apples were green
c the first was red and the second was green
d the first was green and the second was red
e she ate one red and one green apple.
4 Marjut has a carton containing 10 cans of soup. 4 cans are tomato and the rest are
pumpkin. She selects 2 cans at random without looking at the labels.
a Let T represent tomato and P represent pumpkin. Draw a tree
diagram to illustrate this sampling process.
b What is the probability that both cans were tomato soup?
c What is the probability that one can was tomato and the
other can was pumpkin soup?
LARGE SAMPLE SPACES
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Sometimes the number of possible outcomes is sufficiently large that it is a waste of time to
draw a tree diagram. We can still use the same principles we used with the tree diagrams to
calculate the probabilities.
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TREE DIAGRAMS AND BINOMIAL PROBABILITIES (Chapter 20)
Example 5
409
Self Tutor
A hat contains 20 tickets with the numbers 1, 2, 3, ..., 19 and 20 printed on them.
If 3 tickets are drawn from the hat without replacement, determine the probability that
all are prime numbers.
There are 20 numbers of which 8 are primes: f2, 3, 5, 7, 11, 13, 17, 19g
) P(3 primes)
= P(1st drawn is prime and 2nd is prime and 3rd is prime)
8
7
6
£ 19
£ 18
= 20
8 primes out of 20 numbers
7 primes out of 19 numbers after a successful first draw
6 primes out of 18 numbers after two successful draws
¼ 0:0491
EXERCISE 20B.3
1 A lottery has 100 tickets which are placed in a barrel. Two tickets are drawn at random
from the barrel to decide 2 prizes. If John has 2 tickets in the lottery, determine his
probability of winning:
a first prize
b first and second prize
c second prize but not first prize.
d none of the prizes.
2 A bin contains 12 identically shaped chocolates of which 8 are strawberry creams. If 3
chocolates are selected at random from the bin, determine the probability that:
a they are all strawberry creams
b none of them are strawberry creams.
3 A bag contains two white and five red marbles. Three marbles are selected
simultaneously. Determine the probability that:
a all are red
b only two are red
c at least two are red.
SELECTION WITH AND WITHOUT REPLACEMENT
You may have noticed that when we work with tree diagrams, the probabilities of independent
and dependent events are calculated using the same method. The following example compares
two such events.
Consider a box containing 3 red, 2 blue and 1 green marble. Suppose we wish to sample two
marbles:
² with replacement of the first before the second is drawn
² without replacement of the first before the second is drawn.
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Examine how the tree diagrams differ:
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TREE DIAGRAMS AND BINOMIAL PROBABILITIES (Chapter 20)
With replacement
Without replacement
1st
Ey_
R
Wy_
2nd
R
B ( )
Qy_
Ey_
G
R ( )
Ey_
B
Wy_
Ey_
G
R
Qy_
Wy_
B
Qy_
G
Ey_
Wy_
Wy_
B
Qy_
Qy_
G
This branch represents Blue
with the first draw and Red
with the second draw. This
is written as BR.
Notice that:
Wt_
R
Wt_
B ( )
Qt_
G
Et_
R ( )
Qt_
B
B
Qt_
Et_
Wt_
G
² with replacement
P(two reds) =
² without replacement
P(two reds) =
3
6
3
6
£
£
3
6
2
5
=
=
2nd
R
1st
G
R
G
can’t
have
GG
1
4
1
5
We can thus see why replacement is important.
Example 6
Self Tutor
For the example above of the box containing 3 red, 2 blue and 1 green marble, find
the probability of getting a red and a blue:
a with replacement
b without replacement.
a
b
P(a red and a blue)
= P(RB or BR)
=
=
=
3
2
6 £ 6
12
36
1
3
+
2
6
£
P(a red and a blue)
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3
6
= P(RB or BR)
3
2
6 £ 5
12
30
2
5
=
=
=
+
2
6
£
3
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EXERCISE 20B.4
Use tree diagrams to help answer the following questions:
1 Jar A contains 4 blue and 2 red marbles. Jar B contains 1 blue and 5 red marbles. A jar
is randomly selected and one marble is taken from it. Determine the probability that the
marble is blue.
2 Two marbles are drawn in succession from a box containing 2 purple and 5 green marbles.
Determine the probability that the two marbles are different colours if:
a the first is replaced
b the first is not replaced.
3 5 tickets numbered 1, 2, 3, 4 and 5 are placed in a bag. Two are taken from the bag
without replacement. Determine the probability that:
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b both are even
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TREE DIAGRAMS AND BINOMIAL PROBABILITIES (Chapter 20)
4 Jar A contains 3 red and 2 green tickets. Jar B contains 3 red and
7 green tickets. A die has 4 faces marked A and 2 faces marked B.
The die is rolled and the result is used to select either jar A or jar B.
When a jar has been selected, two tickets are randomly selected
from it without replacement. Determine the probability that:
a both are green
411
A
A
B
b they are different in colour.
Example 7
Self Tutor
A bag contains 5 red and 3 blue marbles. Two marbles are drawn simultaneously from
the bag. Determine the probability that at least one is red.
P(at least one red) = P(RR or RB or BR)
draw 2
draw 1
Ru_
R
=
Eu_
B
=
Tu_
R
Wu_
B
R
Ti_
Ei_
=
=
B
5
4
5
8 £ 7 + 8
20+15+15
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50
56
25
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£
3
7
+
3
8
£
5
7
Alternatively, P(at least one red) = 1 ¡ P(no reds) fcomplementary eventsg
= 1 ¡ P(BB) and so on.
5 A bag contains four red and two blue marbles. Three marbles are selected simultaneously.
Determine the probability that:
a all are red
b only two are red
c at least two are red.
6 Box A contains 3 red and 4 green marbles.
Box B contains 5 red and 2 green marbles.
One marble is randomly selected from box A and
A
its colour noted. If it is red, 2 reds are added to
box B. If it is green, 2 greens are added to box B.
B
A marble is then selected from box B.
Find the probability that the marble selected from box B is green.
C
BINOMIAL PROBABILITIES
In many situations there are only two possible outcomes. For example:
² a tennis player either wins or loses a game
² you either catch a bus or you don’t
² you either make a free throw or you miss.
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Binomial experiments are concerned with the repetition of several independent
trials where there are only two possible outcomes, success and failure. Each trial
has the same probability of success.
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