Download Probability - Pearson Schools and FE Colleges

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Key words
Probability
event
outcome
Find the probabilities of events
Understand that if the probability of an event happening is p, the probability of it not
happening is 1 p
An event , such as rolling a dice, can have different outcomes .
When a fair six-sided dice is thrown, the possible outcomes are 1, 2, 3, 4, 5 and 6. Each
outcome is equally likely. The probability of throwing a 2 is 16, as there is a 1 in 6 chance
of this happening.
It is certain that one of the numbers 1 to 6 will be thrown. If an outcome is certain, it
has a probability of 1. So, if the probability of throwing a 2 is 16, the probability of
throwing a number other than 2 is 1 16. We write this as:
P(2) 16
P(not 2) 1 16 56
Example 1
A bag contains discs numbered from 1 to 10. One disc is chosen at random.
What is the probability that the number is:
a) even
b) not even
c) less than 7
d) not less than 7
e) a prime number
f) not a prime number?
Show your answers as a fraction, decimal or percentage.
a)
5
10
0.5
b) 1 21 21 0.5
c)
6
10
e)
4
10
40%
Example 2
3
5
d) 1 6
10
1
4
10
f)
10
10
10
10
6
10
4
10
160 60%
4
10
The prime numbers
between 1 and 10 are 2, 3,
5 and 7.
The probability that Pran is late for school in the morning is 0.1.
a) What is the probability that he is not late for school?
b) How many times would Pran expect to be late in 20 mornings?
a) 1 0.1 0.9
b) 0.1 20 2. Pran would expect to be late twice in 20 mornings.
Exercise 3.1 ..........................................................................................
50 raffle tickets are sold, numbered 1–50. A ticket is chosen at random. What is the
probability that the number is:
a) even
c) not less than 10
e) odd and divisible by 2
Show your anwers as percentages.
26
Maths Connect 2B
b) less than 10
d) odd and divisible by 5
f) not number 6?
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A fair dice is thrown. Copy and complete the table to show the probabilities of each
event. Give your answers as fractions.
Event
Probability of event
happening
Probability of event NOT
happening
Throwing a 6
Throwing an odd number
Throwing a number less than 3
Throwing a prime number
Throwing a number more than 6
A CD contains 15 tracks. The tracks are played at random. What is the probability that
the first track is
a) number 6
b) not number 6?
Copy and colour the spinner so that the probability of it landing on:
a) green is 13, not blue is 56, not white is 12
b) green is 13, not blue is 56, not white is 23.
You need four colours.
The weather forecast predicts that tomorrow there will be a 65% chance of rain. What is
the probability that it will not rain?
There are 20 horses in a field coloured grey, chestnut, bay or black. If a horse is chosen at
random, the probability of bay is 0.5, of grey is 0.25 and of not black is 0.9. How many of
each colour horse are there?
A bag contains 8 sweets. The probability of a caramel being picked
is 0.5. A caramel is removed from the bag and not replaced. A sweet
is chosen at random. What is the probability that it is a caramel?
Using counters may
be helpful.
A school sells 100 raffle tickets numbered 1–100. The winning ticket is drawn at random.
a) Mair has ticket numbers 15, 16, 17, 18, 19. Sean has tickets number 25, 35, 45, 55, 65.
Who has the better chance of winning? Explain your answer.
b) Russell has some tickets. He has a 15% chance of winning. How many tickets does he
have?
The probability of an archer hitting a target with an arrow is 0.9.
Thirty arrows are fired. How many arrows are expected to hit the
target?
Look at Example 2.
Investigation
Kim and Selim have four cards each. Kim has numbers 1, 2, 3, 4 and Selim has
5, 6, 7, 8. They each remove one card, then add together the numbers on their
remaining three cards. Investigate these probabilities for Kim and Selim:
a) the total is an odd number
b) the total is greater than 7.
Probability 27
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Key words
Diagrams and tables
sample space
random
To record all possible outcomes of events systematically
We can show all the possible outcomes of an event on a sample space diagram.
If several outcomes are all equally likely, we say they are random .
Example 1
A coin can show a head (H) or a tail (T). Two coins are thrown. What are the
possible outcomes?
Coin 1 Coin 2
H
H
H
T
T
H
T
T
Example 2
Two tetrahedral dice numbered from 1 to 4 are thrown. The total of the
numbers lying face downwards is found and recorded.
a) List all the possible outcomes.
b) What is the probability that the total of the two dice is:
i) 5
ii) even
iii) less than 4 iv) not less than 4 v) 1.
1
2
3
4
1
2
3
4
5
2
3
4
5
6
3
4
5
6
7
4
5
6
7
8
The sample space is the
shaded numbers.
a) The possible outcomes are 2, 3, 4, 5, 6, 7 and 8.
b) i)
4
16
41
ii)
8
16
21
iii)
3
16
iv) 1 136 1163
v) 0
Exercise 3.2 ..........................................................................................
a) A teacher chooses two pupils from her class. Using B for a boy and G for a girl,
show the four possible outcomes using a sample space diagram.
b) Find the possible outcomes if the teacher chooses three pupils.
Copy and complete the table to show all the possible outcomes when a dice is thrown
and a coin is spun.
Dice
Coin
heads (H)
tails (T)
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Maths Connect 2B
1
1, H
2
3
4
5
6
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Luke has digit cards 1, 3 and 5. Adam has digit
1
3
5
cards 2, 4 and 6. They each choose a card at
2
3
random and add their two numbers together.
4
Copy and complete the sample space diagram to
6
show all of the possible outcomes.
Use it to find the probability that the total will be:
a) 5
b) less than 6 c) not less than 6 d) even.
Use a sample space diagram to show all of the possible outcomes if the numbers are
multiplied together.
Matt has the digit cards 4, 5 and 6.
a) Which 2-digit numbers can he make?
b) What is the probability that a 2-digit number chosen at random is:
i) even
ii) less than 50 ii) not less than 50 iv) divisible by 5?
Maggie has digit cards 1, 3, 4 and 9.
a) How many possible 2-digit numbers can she make?
b) What is the probability that a 2-digit number chosen at random is:
i) odd ii) less than 50 ii) not less than 50 iv) divisible by 3?
Sangheeta and George throw three coins. Sangheeta wins if two or more heads show.
George wins if less than two heads show. Is this a fair game? Explain your answer.
a) Copy and complete the sample space
Spinner A
Spinner A
Spinner B
1
1
2
Spinner B
1 2
diagram to show all possible outcomes
when these two spinners are spun. The
spinner that lands on the highest
number wins. Use ‘A’ to show that
A wins, ‘B’ to show B wins and ‘’ to
show that both numbers were the same.
6
5
3
5
3
1 1
5
4
4
5
1
3
3
6
A
3
3
5
B
5
Does each spinner have the same chance of winning? Explain your answer.
b) Repeat part a) using these two
spinners.
Spinner R
Spinner S
2 4
2 2
5
5
3
4
5
5
4
4
Investigation
Use the idea from Q7 to design two different spinners so that the probability of either
spinner winning is equal.
Diagrams and tables 29
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Key words
Tree diagrams
tree diagram
Draw tree diagrams to show all posssible outcomes
In Lesson 3.2, we saw that the possible outcomes of throwing two coins are:
head, head
head, tail
tail, head
tail, tail
We can also use a diagram called a tree diagram to show all posssible outcomes of
two or more events.
1st coin
2nd coin
outcome
head
head, head
tail
head, tail
head
tail, head
tail
tail, tail
head
tail
Example
An ordinary dice is thrown when a coin is spun. Draw a tree diagram to
show all of the possible outcomes.
dice
1
2
3
4
5
6
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Maths Connect 2B
coin
outcome
head
1, head
tail
1, tail
head
2, head
tail
2, tail
head
3, head
tail
3, tail
head
4, head
tail
4, tail
head
5, head
tail
5, tail
head
6, head
tail
6, tail
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Exercise 3.3 .............................................................................................
Copy and complete the tree diagram
to show all possible outcomes when
these two spinners are spun.
Spinner 1
Spinner
1
Spinner 1
green
Spinner
Spinner
22
Spinner 2
Outcome
red
green, red
blue
green, blue
white
green, white
yellow
black
A football match can finish in a ‘win’, ‘loss’ or ‘draw’. Draw a tree diagram to show all
posssible outcomes when two football matches are played.
Yasmin wants to go to two clubs during the week. On Monday she can choose
swimming (S), kick boxing (K), guitar (G) or tai chi (T). On Wednesday she can choose
chess (C) or rock climbing (R). Draw a tree diagram to show all the possible combinations
of clubs.
A restaurant has a choice of potatoes and vegetables
with lunch. Draw a tree diagram to show possible
choices of one type of potato and one vegetable.
Bag A contains a red and a blue counter. Bag B contains a green, a black, a white and a
yellow counter. Draw a tree diagram to show all the possible outcomes when a counter is
taken at random from each bag.
A bag contains three cubes: a blue, a yellow and a white. A cube is removed from the bag,
its colour recorded and then it is replaced. Another cube is removed from the bag. Draw a
tree diagram to show all possible outcomes of the colours of the two cubes removed from
the bag.
Investigation
Look at the tree diagrams you have drawn. Count the total number of outcomes for
each one. What is the connection between the number of possible outcomes for each
individual event and the total number of outcomes for both events? Use your answer
to predict the total number of outcomes when a coin is spun and a 4-sided dice,
numbered 1–4 is thrown. Draw the tree diagram to check if your prediction is correct.
Tree diagrams 31
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Key words
Estimating probability
theoretical probability
estimated probability
To estimate probability from experiments
For a fair, six-sided dice, the theoretical probability of throwing a six is 16. We find
the theoretical probability by using the possible outcomes.
For a particular six-sided dice, we could find the estimated probability by
experimenting. This might be different from the theoretical probability.
The estimated probability must be between 0 and 1 inclusive and written as a fraction,
decimal or percentage. If a coin is thrown 50 times and 28 tails are recorded, the
estimated probability of the coin landing on tails is 2580, 0.56 or 56%.
Example
Three coins are thrown repeatedly and the number of heads each time is
recorded.
Number of heads
Frequency
0
1
2
3
12
14
15
9
a) How many times were the coins thrown?
b) Find the estimated probability for each number of heads thrown.
c) Use the answers to part b) to estimate the probability
of three coins showing two or more heads.
12 14 15 9 50
a) 50 times
b)
Number of heads
0
Estimated probability
12
50
1
0.24
14
50
2
0.28
15
50
3
0.3
9
50
0.18
c) 0.3 0.18 0.48
Exercise 3.4 ..........................................................................................
A dice is thrown and the number showing is recorded.
Number
Frequency
1
2
3
4
5
6
12
8
7
9
14
10
a) How many times was the dice thrown?
b) Find the estimated probability for each of the six numbers.
c) What is the estimated probability that the dice will show:
i) An even number
ii) A number less than 4
iii) Not a 2?
Rick shuffles a pack of postcards, turns the top card over and records the type of
picture in a frequency table.
Type of card
View
Animal picture
Transport picture
Frequency
5
21
14
Find the estimated probability for each type of card. Show your answers as decimals.
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Maths Connect 2B
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A class count the number of paper clips in 30 small boxes. They find that 26 boxes
contain 51 or less paper clips. What is the estimated probability that the next box counted
contains more than 51 paper clips?
To win a prize at tombola, Jane has to pick from a drum a ticket ending in 0 or 5.
Assuming that all ticket numbers from 1–500 are put in the drum at the start, what is the
estimated probability of Jane winning a prize?
Work with a partner. You will need dominoes.
Share the dominoes equally and then each select
a domino at random. Is there a ‘snap’ between
any of the numbers? (Do not count a ‘double’ as a snap.)
Repeat this experiment 50 times, then record your results in a table like this.
Snap
Snap!
No snap
Tally
Frequency
Estimated probability
This activity requires three dice.
a) What is the lowest possible total when three dice are thrown together? The highest total?
b) Use your answers to part a) to complete a tally chart showing
Total
all of the possible totals when three dice are thrown.
c) Throw the three dice a total of 50 times. Each time record the Tally
total in your table.
Frequency
d) Find the estimated probabilities for each of the possible totals.
Estimated probability
This activity requires a dice.
Throw a dice, adding up the numbers thrown until you reach a total of six or more.
Record the number of throws needed to get this total in the table. You do not need to
record your total. Repeat this experiment 40 times.
Number of throws for 1
a total of 6 or more
2
3
4
5
6
Tally
Frequency
Estimated probability
Complete the table by finding the estimated probabilities for each amount of throws.
This activity requires the spinner from worksheet 5.4, and a cocktail stick.
Use the cocktail stick to make a hole in the spinner. Do not put the hole in the centre of
the spinner. (The idea is to make the spinner biased.) Spin your spinner 60 times,
recording the number it lands on. Copy this table and use it to record your results.
Number
1
2
3
4
5
6
Tally
Frequency
Estimated probability
Comment on your results.
Estimating probability 33
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Key words
Better estimates of probability
If an experiment is repeated there will usually be different results
Increasing the number of times an experiment is performed will usually
give a better estimate of probability
random
unpredictable
reliable
A dice is thrown six times with these results:
The number thrown is random or unpredictable .
We would not say that is impossible to throw a 6 based only on these throws as we
have not thrown the dice enough times.
When doing an experiment, in general, the more times we perform the experiment the
more reliable our results will be.
Also, if we throw the dice another 6 times we do not expect to get exactly the same
results each time.
Example
Helen throws a coin 10 times and counts
the number of heads and tails.
Number of tails
6
Number of heads
4
She throws the coin a further 50 times
with these results:
Number of tails
23
Number of heads
27
a) Calculate the estimated probabilities for both frequency tables.
b) Draw a bar-line graph showing these probabilities.
a)
Estimated probability
Number of tails
6
0.6
Number of heads
4
0.4
Estimated probability
Number of tails
23
0.46
Number of heads
27
0.54
Estimated probabilities of a
coin showing heads or tails
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10 s a
N
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50 s a
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0.7
0.6
0.5
0.4
0.3
0.2
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Estimated
probability
b)
34
Maths Connect 2B
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Exercise 3.5 .............................................................................................
This activity requires two dice.
Throw the two dice 10 times, recording the total for each throw in the table below.
Total
2
3
4
5
6
7
8
9
10
11
12
Tally
Frequency
Draw a bar-line graph for your results, plotting the total on the
x-axis and the frequency on the y-axis. Repeat this experiment
for a total of 20 throws, and then for a total of 50 throws. Each
time draw a bar-line graph.
Compare the shape of your three graphs and write a sentence
about your results.
You can use your first 10
throws as part of your 20,
and your 20 as part of
your 50.
This activity requires a coin.
Look at the Example. Do Helen’s experiment for yourself. Record your results for 10, 30
and 50 throws of the coin. Draw a bar-line graph to show your results.
This activity requires ‘Lego’ bricks.
What is the probability of a piece of ‘Lego’
landing ‘face up’, ‘face down’ or ‘on one of its
sides’?
Decide what information you are going to record and draw a table. Throw the ‘Lego’ 10
times, 30 times and 50 times. For each set of throws, find the estimated probability of it
landing on a particular side.
Repeat this activity using a different sized piece of ‘Lego’.
Compare your results. Do different pieces of ‘Lego’ have different probabilities for
landing in a particular way?
This activity requires a calculator with a ‘random’ button.
To simulate throwing an ordinary dice:
● Use the ‘random’ button to get a decimal number between 0 and 1.
● Multiply this by 6, then add 1.
● Use the number before the decimal point to simulate the number on the dice.
For example: 0.632 6 1 4.792. This is the same as throwing a 4 on a dice.
Use this method to generate 60 ‘throws’ of a dice, recording your results in a table.
Random number
1
2
3
4
5
6
1
6
1
6
1
6
1
6
1
6
1
6
Tally
Frequency
Theoretical frequency
Compare your results to the theoretical frequencies of throwing numbers 1–6 on an
ordinary dice.
Better estimates of probability 35
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Key words
Comparing probabilities
theoretical probability
estimated probability
Compare theoretical with experimental probabilities
Throwing a dice produces random outcomes. We can find the theoretical probability
of throwing a 6 and also, by throwing the dice, find the estimated probability . We do
not always expect these two values to be the same.
Example
Three coins are thrown and the number of tails counted.
Copy and complete the table to show:
a) the possible outcomes b) the theoretical probability for each outcome.
Number of tails
0
a)
Outcome
b)
Theoretical probability
1
2
3
Caitlan throws three coins 100 times. Here are her results:
Number of tails
Frequency
0
1
2
3
12
39
35
14
c) Estimate the probability for each outcome.
d) Compare the theoretical and estimated probabilities.
Number of tails
0
1
2
3
a)
Outcome
HHH
HHT
HTH
THH
TTH
THT
HTT
TTT
b)
Theoretical probability
1
8
Estimated probability
12
100
c)
0.125
0.12
3
8
0.375
39
100
0.39
3
8
0.375
35
100
0.35
1
8
0.125
14
100
0.14
d) The theoretical and estimated probabilities were quite close, but we would not expect
them to be exactly the same. Increasing the number of times the experiment was
performed may improve the accuracy of the estimated probabilities.
Exercise 3.6 ..........................................................................................
Four pupils each throw two dice. They record the number
of times the two numbers are the same.
Name
36
Number of throws
Both dice same
Both dice different
Falek
30
4
26
Paolo
120
17
103
Alan
70
13
57
Nai
180
22
158
Maths Connect 2B
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a) Which pupil’s results are most likely to give the best estimates of the probabilities?
Explain your answer.
b) Complete the table to show the results of all 400 throws.
Both dice same
Both dice different
Number of throws
Estimate of probability
c) Draw a table to show all the possible outcomes when two dice are
thrown. Use ‘S’ to show that the dice are the same, ‘D’ to show they
are different.
1
2
1
S
D
2
d) What is the theoretical probability that the two dice:
i) show the same number?
ii) show different numbers?
e) Use the theoretical probabilities to calculate for 400 throws the number of times the
dice will be:
iii) the same
iv) different.
f) Compare the pupils’ results with the theoretical results. Why are they not the same?
a) This activity is for two and requires dice and
Player A
Player B
counters. Each player takes turns to throw two
Prime
Single
dice, adding the scores together. If the total is one
Even
number Odd
digit
of their four categories of outcomes, a counter is
placed on that square. The first player to cover all
Factor Triangle
Square Multiple
of 8 number
number
of 5
four of their squares wins.
b) Draw a sample space diagram to show all the
possible outcomes when two dice are thrown and the scores added together.
Use the theoretical probabilities to show that the game in part a) is not a fair game.
c) Design a similar game where both players have an equal chance of winning.
This activity is for two players and requires counters and a bag.
One pupil chooses 20 counters of a mixture of blue and red colours and places them in a
bag. The other pupil takes a counter from the bag, notes the colour and replaces it.
Repeat 10 times. Estimate the number of each colour counter. Repeat the experiment a
total of 30 times, then 50 times. Record your results in a table.
Number of counters taken from the bag
10
30
50
Number of red counters
Estimated probability
Number of blue counters
Estimated probability
Empty the bag to see how close your estimates were.
Comparing probabilities 37