Download Probability - Pearson Schools and FE Colleges

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Key words
Probability
Find the probabilities of events
Know that if the probability of an event is p, the probability of it not
happening is 1 p
event
outcome
theoretical
probability
The event of throwing a standard six-sided dice will give an outcome of 1, 2, 3, 4, 5
or 6 with each outcome having a theoretical probability of 61. We can write ‘the
probability of a 1 is 61’ or P(1) 16.
Provided all possible outcomes are equally likely, the theoretical probability of an
the number of ways the outcome can happen
outcome the number of possible outcomes
We say that P(1) 16 is the theoretical probability, but the probability that is worked
out from an experiment (that is, by throwing a dice a number of times) may be
different from this.
Probability of an outcome happening 1 probability of the outcome not happening.
Example 1
Nova buys two packets of sweets. Packet 1 has 8 fruit chews and 4 fizz
bombs. Packet 2 has 9 fruit chews and 6 fizz bombs. From which packet
should she choose a sweet at random to have the greater probability of
selecting a fruit chew?
P(choosing a fruit chew from packet 1) 182 6400
P(choosing a fruit chew from packet 2) 9
15
36
60
She should choose from packet 1.
Example 2
Route 1:
Natacha has a choice of two bus routes for her journey to work. She thinks
that the probability of her being late if she uses Route 1 is 17 and if she uses
Route 2 the probability is 18%. Calculate the probability of her not being late
for each route. Which route should she use?
P(not being late) 1 71
77 71
67
Route 2: P(not being late) 100% 18%
82%
6
7
86% (to two decimal places). There is a higher probability of
her not being late if she uses Route 1.
26
Change the probabilities
into fractions with a
common denominator.
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Exercise 3.1 .............................................................................................
Jem has two boxes of football cards. Box 1 has 17 Premiership cards and 10 Division 1
cards. Box 2 has 11 Premiership cards and 7 Division 1 cards. Jem decides to pick a card
at random. From which box should Jem choose if she wants to pick a Premiership card?
The weather forecast predicts the probability of rain for the next day as 45%. What is the
probability of it not raining?
A bag contains 20 coloured counters: blue, green, yellow and red.
P(blue) 0.5, P(yellow) 0.25, P(not green) 0.9. How many of each colour counter are
there in the bag?
A bag contains 20 cubes with P(blue) 0.4. A blue cube is taken from the bag and not
replaced. Another cube is taken at random from the bag. What is the probability that it is
also blue?
Copy and complete the spinner using four colours
(blue, yellow, white and red), so that the probability
of it landing on:
not blue is 68 AND
not yellow is 58 AND
not blue or white is 58.
George and Neil choose some numbers to play a lottery. They can choose five numbers
between 1 and 80.
George chooses numbers 2, 3, 4, 5 and 6.
Neil chooses numbers 17, 23, 31, 40 and 56 as he believes that as there are more two-digit
numbers than single-digit numbers, he will have a better chance of winning.
Is Neil correct? Explain your answer.
A traffic light has four possible combinations of lights: red; green; amber; red and amber.
The probability of the lights showing green is 0.6, amber is 0.05,
and of red and amber is 0.01.
a) What is the probability of the traffic lights showing
just red?
b) If it takes 2 minutes for the lights to complete a full cycle,
how long do each of the four different combinations
of lights show for?
Change 2 minutes
to seconds.
Investigation
What is the probability that a number between 100 and 999 inclusive, chosen at
random, is:
a) divisible by 2?
d) divisible by 9?
b) divisible by 4?
e) not divisible by 9?
c) divisible by 5?
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Key words
Possible outcomes
mutually
exclusive
sample space
Know that the sum of all the mutually exclusive outcomes of an
experiment add up to 1
Find all possible outcomes of an experiment
Two outcomes are mutually exclusive if only one of them can happen at any one
time. For example, when a dice is thrown, the outcomes are 1, 2, 3, 4, 5 or 6. Any two
outcomes cannot happen at the same time, so the outcomes 1 and 2 are mutually
exclusive. All the mutually exclusive outcomes of an event add up to 1. So, for
throwing a dice:
P(1) P(2) P(3) P(4) P(5) P(6) 16 16 16 16 16 16 1
When finding all of the possible outcomes of an event it is often helpful to use a table
or a sample space diagram. Alternatively, work systematically (using an order) so
no outcomes are missed.
Example
a) Three coins are tossed. Write all the possible outcomes in a systematic way.
b) What is the probability that when three coins are tossed:
i) you will get one or more tails?
ii) you will get less than two heads?
a) The possible outcomes are:
head, head, head
(3 heads)
head, head, tail
head, tail,
tail,
head
head, head
head, tail,
tail
tail,
head, tail
tail,
tail,
head
tail,
tail,
tail
b) i)
ii)
7
8
4
8
(2 heads)
All the possible outcomes are
known as the sample space.
There are three possible ways
of getting two heads and one
tail, because of the order in
which they occur.
1
2
(1 head)
(0 heads)
There are seven different
outcomes with one or more
tail out of a total of eight.
Less than two heads is either one
head or zero heads, so 38 18.
Exercise 3.2 .............................................................................................
a) In how many different orders can a total of 80p be put into the ticket machine in a car
park, using a 50p, a 20p and a 10p coin?
b) If the coins are put into the machine in a random order, what is the probability that
the 10p will be put in before the 20p?
Use the Example to help
Write down, in a systematic way, the outcomes when four
you.
coins are tossed.
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Five friends, Ananth, Ben, Carlo, Davina and Erica, have
four tickets for a concert between them.
Ananth must have a ticket as she is driving the others. How
many different groups of four people can go to the concert?
Ananth, Ben, Carlo and
Davina going is the same
as Ananth, Ben, Davina
and Carlo going.
Another group of five friends, Faruk, Gray, Helen, Ian
and Jaymin, have four tickets for the concert.
If Helen goes to the concert, so must Ian. How many
different groups of four people can go to the concert?
The probabilities of a bus being either early or
late at two towns are shown in the table.
a) Copy and complete the table to show the
probabilities of a bus being on time.
Town
Early
On time
Late
Northtown
2.5%
17.5%
Southtown
3.6%
22.8%
b) Copy and complete the table showing the expected number of buses early, on time or
late for each town.
Number of
buses each
week
Northtown
40
Southtown
250
Number of
Number of
Number of
buses expected buses expected buses expected
early
on time
late
Fleur buys three different boxes of chocolates.
Complete the table showing the probability of
choosing a type of chocolate at random from
each box.
Dark
Brambletons
0.15
Dairy Cream
Milk box
Milk
0.25
30%
1
2
White
5%
2
5
Each box contains 20 chocolates. How many of each type of chocolate are in each box?
A bag contains three different colours of marbles. The probability of choosing a red
marble at random is 112 times the probability of choosing a blue marble. The probability
of choosing a green is 16.
a) What is the probabilities of choosing: i) a red marble ii) a blue marble?
b) The bag contains 18 marbles. How many of each colour are in the bag?
Investigation
How many ways can two boys and two girls sit on four chairs arranged in a line so
that the order is different each time?
What is the probability that a seating order chosen at random has:
a) the two girls sitting next to each other? b) boys and girls sitting alternately?
Possible outcomes 29
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Key words
Tree diagrams
tree diagram
Draw tree diagrams to show all possible outcomes for two or more events
A tree diagram is used to show all the possible outcomes for two or more events.
Example 1
a)
a) Draw a tree diagram to show the outcomes when three coins are tossed.
b) What is the probability that three coins will show:
i) one tail only ii) less than three tails?
Coin 1
Coin 2
Coin 3
head
head
tail
head, head, head
head, head, tail
tail
head
tail
head, tail, head
head, tail, tail
head
head
tail
tail, head, head
tail, head, tail
tail
head
tail
tail, tail, head
tail, tail, tail
head
tail
b) i)
ii)
3
8
7
8
Example 2
Head, head, tail;
head, tail, head;
tail, head, head
are the outcomes that
include one tail, out of a total
of eight possible outcomes.
P(0 tails) P(1 tail) P(2 tails)
81 38 38
Andy, Beth and Carlo get on a bus. There are only two seats left. Draw a tree
diagram to show which two of the three get a seat, and work out the
probability of Andy and Beth having seats.
1st seat
Andy
Beth
Carlo
2nd seat
Beth
Maths Connect 2R
Outcome
Andy and Beth
Carlo
Andy and Carlo
Andy
Beth and Andy
Carlo
Beth and Carlo
Andy
Carlo and Andy
Beth
Carlo and Beth
P(Andy and Beth) 62 31
30
Outcome
Andy and Beth getting a seat
is the same as Beth and Andy
getting a seat.
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Exercise 3.3 .............................................................................................
A football match can finish in a ‘win’, ‘lose’ or ‘draw’.
a) Draw a tree diagram to show all possible outcomes when two football matches are
played.
b) Draw a tree diagram to show all possible outcomes when three football matches are
played. What is the probability that all three matches end in a draw, if each outcome
has the same probability of happening?
Each lunchtime a restaurant offers a ‘three-course
Starter
Main course
Dessert
meal deal’ with a choice of starter, main course
soup
fish & chips
ice cream
and dessert.
melon
pasta
treacle tart
a) Draw a tree diagram showing the available
cheese salad fruit salad
choices for a three-course meal.
b) How many days can a person eat at the restaurant and eat a different meal each day?
a) A bag contains three cubes coloured blue, yellow and white. A cube is removed from
the bag, its colour is recorded and then it is replaced. Another cube is removed from
the bag and its colour is recorded. Draw a tree diagram to show all possible outcomes.
b) Draw a tree diagram showing all the possible outcomes when the first cube is
removed from the bag, it is not replaced and a second cube is chosen.
A game requires contestants to answer ‘true’ or ‘false’ to a question.
If a contestant does not know the answer there is an equal chance
that she will guess either ‘true’ or ‘false’.
Amanda answers four questions, guessing her answer each time.
a) Draw a tree diagram to show all possible sets of four answers.
b) The correct answers to Amanda’s four questions are ‘true’, ‘true’, ‘false’, ‘true’.
What is the probability that Amanda guessed correctly?
A bag contains three marbles coloured blue, red and yellow. A marble is removed from
the bag, its colour is noted and then it is placed into a second bag containing a white and
black marble. A marble is then taken from the second bag. Draw a tree diagram showing
the possible outcomes for the two marbles that are removed.
a) Use the tree diagram in Example 1 to find the probability of three coins showing:
i) 0 heads
ii) one head
iii) two heads
iv) three heads
v) no tails
vi) one tail
vii) two tails
viii) three tails
ix) more than one head
x) less than two tails.
b) Use your answers to match up a statement from column 1 with a statement from
column 2.
Column 1
1 P(0 heads)
P(0 heads)
P(1 head)
P(2 heads)
P(1 head)
1 P(3 tails)
P(3 heads)
Column 2
P(2 tails)
P(2 tails)
P(1 head) P(2 heads) P(3 heads)
P(0 tails)
P(3 tails)
P(3 tails)
P(1 tail)
For example,
P(2 heads) P(1 tail)
because if two out of the three
coins show heads, the other
coin must show a tail.
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Key words
Estimating probability
estimated probability
theoretical probability
Estimate probability from experiments
When an experiment is performed the results can be used to find the
estimated probability of a particular outcome or set of outcomes. This estimated
probability might be different from the theoretical probability .
The probability must be between 0 and 1 inclusive and written as a fraction or decimal.
It can also be written as a percentage.
For example, if a fair coin is thrown 50 times with tails showing 23 times, the
estimated probability of a tail is 2530 or 0.46 or 46%. The theoretical probability of a tail is
1, or 0.5 or 50%.
2
Example
A dice is thrown 60 times and the number showing is recorded.
Number
1
2
3
4
5
6
Frequency
9
11
10
12
9
9
a) Find the estimated probability for each of the six numbers. Show your
answers as a decimal correct to two decimal places and a fraction in its
lowest form.
b) What is the estimated probability that the dice shows:
i) an even number
ii) a number greater than 4 iii) not a 6?
c) Another dice is thrown 60 times. The probability of it showing a 6 is
estimated as 0.22 to two decimal places. How many times did the dice
show a 6?
a)
Number on dice
1
2
3
4
5
6
Estimated probability (decimal) 0.15
0.18
0.17
0.2
0.15
0.15
How many times did the
each
9 coin
3 show
11
10 of1 the
12 two
1 sides?
9
3 9
3
Estimated probability (fraction) 60 20 60
60
6 60
5 60
20 6 0
20
b) i)
11
60
6120 690 6302 185
c) 0.22 60 13.2
ii)
3
20
230 260 130
iii) 1 230 2170
so 13 times
Exercise 3.4 ..........................................................................................
Sian is playing darts. She records the type of score for 80 darts in a frequency table.
Type of score
Miss
Single
Double
Treble
Bullseye
Frequency
6
45
18
8
3
Find the estimated probability for each type of score. Show your answers as percentages.
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The contents of 47 boxes of drawing pins are counted. 36 boxes contain 52 or less
drawing pins. What is the probability that the next box counted contains more than 52
drawing pins?
A dice is thrown 60 times. The estimated probabilities for each of the six outcomes are
shown in the table. Each probability is rounded to two decimal places.
Number showing on dice
1
2
3
4
5
6
Estimated probability
0.15
0.20
0.13
0.17
0.18
0.17
Calculate the frequency of each number showing.
Requires a dice. Throw a standard six-sided dice. Throw it again. Add the two numbers
showing and repeat until a total of six or more is reached.
What is the lowest number
a) Draw a table to record the number of throws taken for
of throws possible for a
this experiment.
total of six? What is the
b) Perform this experiment 50 times, recording the number
greatest number?
of throws on your data collection sheet.
c) Estimate the probability for each of the number of throws taken to reach a total of six
in your table.
Requires three dice. In a fairground game, three dice are thrown.
The player wins if two or three dice show the same number.
Otherwise the player loses.
a) Copy the table below. Throw the three dice 50 times,
recording the outcomes.
Three dice
the same
Two dice
the same
No dice
the same
Lucky Dice!!
10p a go
Win 20p if 3 dice are the same.
Your money back if 2 dice are the same.
Lose if no dice are the same.
Tally
Frequency
Estimated
probability
How much would you have paid to play 50 times?
How much would you have won getting three dice the same?
How much money would you get back getting two dice the same?
Would you make a profit or a loss on this game?
b)
c)
d)
e)
Frequency 20p
Frequency 10p
Requires digit cards numbered 1 to 9. Shuffle the cards. Turn each card over in turn. Stop
when the value of the card turned over is smaller than the last.
For example,
3
5
7
2
This shows that four cards were turned over so a tally mark
is entered under the ‘4’ in the data collection sheet.
Draw a data collection sheet with these headings: ‘Number of cards turned’, ‘Tally’,
‘Frequency’, ‘Estimated probability’. Carry out this experiment 30 times, recording the
number of cards turned over each time.
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Key words
Better estimates of probability
bias
Understand that if an experiment is repeated there will usually be
different results
Understand that increasing the number of times an experiment is
performed will usually give a better estimate of probability
A fair, six-sided dice is thrown six times with these results: 1, 5, 3, 1, 2, 4.
If we based our experimental probability only on these throws, we might believe it is
impossible to throw a 6. However, it is likely that we have got this experimental
probability because the dice has only been thrown a small number of times.
In general, when doing an experiment, the more times we perform it, the closer our
results will be to the theoretical probabilities.
If we throw the dice another six times we do not expect to get exactly the same results
each time.
If, after throwing the dice 50 or 60 times, we still had not thrown a 6, we might
conclude that the dice is biased (that is, not fair).
Example
Freya throws a coin ten times recording the number of heads and tails. She
throws the coin a further 40 times, then another 50 times with these results:
Total number of throws
10
50
100
Number of tails
6
28
49
Number of heads
4
22
51
a) Calculate the estimated probabilities.
b) Draw a bar-line graph showing these probabilities. Comment on Freya’s
results.
a)
Total number of throws
10
Estimated
probabilities
50
Estimated
probabilities
100 Estimated
probabilities
Number of tails
6
0.6
28
0.56
49
0.49
Number of heads
4
0.4
22
0.44
51
0.51
b)
Estimated probability
0.7
0.6
0.5
Number of tails
0.4
Number of heads
0.3
0.2
0.1
0
10 throws 50 throws 100 throws
Number of throws
The graph clearly shows that the more times the experiment was carried out, the closer the
experimental probabilities of heads and tails became to 21.
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Exercise 3.5 .............................................................................................
Work in pairs.
Choose a total of 20 blue and red counters and place them in a bag. Ask your partner to
take a counter from the bag, note the colour and replace it. Repeat ten times. Estimate the
number of each colour counter in the bag. Repeat the experiment a total of 30 times, then
50 times. Record your results in a table with these headings: ‘Number of counters taken
from bag’, ‘Number of red counters’, ‘Estimated probability’, ‘Number of blue counters’,
‘Estimated probability’.
Draw a graph to illustrate your results.
Compare the shape of your three graphs and write a sentence about these results.
Empty your bag. Did the accuracy of your results improve as you increased the number
of times the experiment was repeated?
Requires a calculator with a ‘random’ button to simulate throwing a four-sided dice.
Use the ‘random’ button to get a decimal number between 0 and 1.
Multiply this by 4 then add 1.
Use the number before the decimal point to simulate the number on the dice.
For example: 0.632 4 1 3.528. This is the same as throwing a 3 on a dice.
Generate 60 ‘throws’ of a four-sided dice, recording your results in a table.
Random number
1
2
3
4
Tally
Frequency
Theoretical frequency
Compare your results to the theoretical frequencies. How could you use your calculator
to simulate throwing a coin and recording if it landed on heads or tails?
Work in pairs. Requires cocktail sticks or matchsticks, card and regular shape templates.
Design a regular shaped spinner with each section of the spinner numbered differently. It
is important that your partner does not know the number of sides your spinner has, or
the numbers written on it.
Spin your spinner where your partner cannot see it, telling him or her what number it
lands on each time so it can be recorded. Repeat this 20 times. Your partner should now
guess the number of sides your spinner has, and how it is numbered.
Continue until you have spun your spinner 40 times. Again your partner should make a
guess about your spinner.
Spin your spinner a total of 60 times with your partner making a final guess.
Did the accuracy of the guesses improve the more times the spinner was spun?
Better estimates of probability 35
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Key words
Theoretical and estimated probability
random
theoretical
estimate
Compare theoretical with estimated probabilities from an experiment
Throwing a dice is a random event. We can find the theoretical probability of
throwing a six. Then, by throwing the dice, we can find an estimate of this
probability. We do not expect these two values to be the same.
Example
Two dice are thrown and the total of the dice recorded.
Copy and complete the table.
Total on dice
2
3
a)
Possible outcome
1, 1
b)
Theoretical probability
4
5
6
7
8
9
10
11
12
Terry throws two dice 100 times, recording the total score.
Total on dice
2
3
4
5
6
7
8
9
10
11
12
Frequency
3
5
9
10
13
19
9
12
11
7
2
c) Estimate the probability for each outcome.
d) Compare the theoretical and estimated probabilities.
Total on dice
2
3
4
5
6
7
8
9
10
a)
Possible outcome
1, 1
1, 2
2, 1
1, 3
3, 1
2, 2
1, 4
4, 1
2, 3
3, 2
1, 5
5, 1
2, 4
4, 2
3, 3
1, 6
6, 1
2, 5
5, 2
3, 4
4, 3
2, 6
6, 2
3, 5
5, 3
4, 4
3, 6
6, 3
4, 5
5, 4
4, 6 5, 6 6, 6
6, 4 6, 5
5, 5
b)
Theoretical probability 0.03 0.06 0.08 0.11 0.14 0.17 0.14 0.11 0.08 0.06 0.03
c)
Estimated probability
0.03 0.05 0.09 0.1
The theoretical probability
is 316 0.03 (to 2 d.p.).
11
12
0.13 0.19 0.09 0.12 0.11 0.07 0.02
The theoretical probabilities do
not total 1 because of rounding.
d) The theoretical and estimated probabilities were quite close, but we would not expect them to be
exactly the same. If we increase the number of times the dice is thrown, the theoretical and
estimated probabilities are likely to get closer.
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Exercise 3.6 .............................................................................................
Four pupils each throw two dice and record how many times the numbers are the same:
Name
Number of throws
Dice same
Dice different
Falek
30
4
26
Paolo
120
17
103
Alan
70
9
61
Nai
180
32
148
a) Which pupil’s results are most likely to give the best estimates of the probabilities?
Explain your answer.
b) Copy and complete the table to show the results of all 400 throws.
Dice same
Dice different
Number of throws
Estimated probability
c) Draw a sample space diagram showing all 36 outcomes when two dice are thrown.
Shade in the squares that show the two dice having the same score.
What is the theoretical probability that the two dice:
Draw a 7 7 table.
i) show the same number ii) show different numbers?
d) Use these probabilities to calculate for 400 throws the number of times the dice will be:
i) the same ii) different.
e) Compare the pupils’ results with the theoretical results. Why are they not the same?
A game for two players: Requires two dice and five counters of two different colours.
Copy the table. Take it in turns to choose a number between 2 and 12, writing the chosen
number in a space on your table.
Player 1
Player 2
Take turns to throw the two dice. If the total of the two dice matches your chosen number
then place a counter on that square.
The object of this game is to be the first player to cover his/her chosen numbers. Once
the game has been won, use the theoretical probabilities from the Example to calculate if
the game was fair. Play the game again. Does the player who goes first have an
advantage or can the two players choose the numbers so that the game is always ‘fair’?
Requires cocktail sticks/matchsticks, card, scissors and angle measurer. Design a circular
spinner, using your angle measurer to make sections of different sizes.
Calculate the theoretical probabilities for each of the sections, and
then use your spinner to calculate the experimental probabilities.
You will need to design a data collection sheet and decide the
number of times you will perform the experiment.
Compare your theoretical and experimental probabilities.
90°
70° 20°
50° 130°
Theoretical and estimated probabiliity 37