Probability Grades F to A
Probability scale and chance
Mutually exclusive
Listing outcomes
Hyperlinks!
Finding a probability
Experimental and theoretical probability
Relative frequency
Multiple events
Tree diagrams
Tree diagrams and dependent events
SUCCESS CRITERA: WHERE ARE WE NOW? Level F2 F1 E3 E3 C3 C2 B2
B1 A3
Learning outcomes: I can put probabilities on a probability scale. I can list all the outcomes from a certain event. I can calculate the probability of an event occurring, know that the probability of all events adds to 1. I can calculate with mutually exclusive probabilities. I can use both experimental and theoretical probability. I can use relative frequency to estimate probabilities. I can calculate the probability of multiple independent events. I can use a tree diagram to solve probability problems I can use a tree diagram with compound or dependent events.
Probability scale and chance Lesson Objective: Can I describe chance and use a probability scale? Grade F
What’s the difference between chance and probability? Chance is described in words. Eg. Impossible, Unlikely, Evens, Likely, Certain Probability is given a numerical value. Probability is given as a fraction, decimal or percentage.
What is a probability scale? This is a line where events can be placed to show the chance or probability of an event occurring. Impossible
Unlikely
Evens
Likely
Certain
0
1 4
1 2
3 4
1
Chance
Probability
Putting events on the probability scale: Chance events (eg. Winning the lottery, eating chips) can be placed in different positions depending on who is doing the placing. Probability events must be put in a certain place.
Placing probability events on the scale: A - P(flipping a head on a coin) B - P(rolling a number under 7 on a dice) I need to split the scale C - P(rolling a 5 on a dice) up into 6 equal bits as
C 0
1 6
One number out of six is a head
there are 6 sides on a dice
A 2 6
One side out of two is a head
1 2
B
4 6
5 6
1
All the numbers on a dice are less than 7
Your turn: A - P(flipping a head or a tail on a coin) B - P(rolling an odd number on a dice) C - P(rolling a number less than 5 on a dice)
0
1 6
2 6
B
C
1 2
4 6
A 5 6
1
SUCCESS CRITERA: WHERE ARE WE NOW?
Level F2 F1 E3 E3 C3 C2 B2 B1 A3
Learning outcomes: I can put probabilities on a probability scale. I can list all the outcomes from a certain event. I can calculate the probability of an event occurring, know that the probability of all events adds to 1. I can calculate with mutually exclusive probabilities. I can use both experimental and theoretical probability. I can use relative frequency to estimate probabilities. I can calculate the probability of multiple independent events. I can use a tree diagram to solve probability problems I can use a tree diagram with compound or dependent events.
R A G
Listing outcomes Lesson Objective: Can I list all possible outcomes of an events or events? Grade F
What does this mean? You should be able to write down all the possible outcomes that could happen. Make sure you do this in a logical way. This will mean that you don’t miss any outcomes.
An example of listing outcomes: A meal deal at a shop means that you can choose from 3 different sandwiches (ham, cheese and tuna) and 3 different drinks (orange, apple and water). List all the possible choices that could be made. Write down the sandwich types and then the drinks that are available: Ham Orange Apple Water
Cheese Orange Apple Water
Tuna Orange Apple Water
Answer: 9 possible outcomes
One for you to have a go at: A dice (6 sides) is rolled and a coin is flipped at the same time. Write all the possible outcomes that there could be – how many are there?. Heads & 1, Heads & 2, Heads & 3, Heads & 4, Heads & 5, Heads & 6 Tails & 1, Tails & 2, Tails & 3, Tails & 4, Tails & 5, Tails & 6
Answer: 12 outcomes
SUCCESS CRITERA: WHERE ARE WE NOW? Level F2 F1 E3 E3 C3 C2 B2
B1 A3
Learning outcomes: I can put probabilities on a probability scale. I can list all the outcomes from a certain event. I can calculate the probability of an event occurring, know that the probability of all events adds to 1. I can calculate with mutually exclusive probabilities. I can use both experimental and theoretical probability. I can use relative frequency to estimate probabilities. I can calculate the probability of multiple independent events. I can use a tree diagram to solve probability problems I can use a tree diagram with compound or dependent events.
Finding the probability of an event Lesson Objective: Can I find the probability of an event? Grade E
You need to remember this: 𝐻𝑜𝑤 𝑚𝑎𝑛𝑦 𝑤𝑎𝑦𝑠 𝑡ℎ𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒 𝑦𝑜𝑢 𝑤𝑎𝑛𝑡 𝑐𝑎𝑛 ℎ𝑎𝑝𝑝𝑒𝑛 𝑇𝑜𝑡𝑎𝑙 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 It is often easier to give a probability as a fraction using the rule above. If you are given probabilities in decimals, then use decimals. Avoid percentages, although they aren’t wrong.
Here’s how the rule works: A bag contains 6 blue marbles, 5 red marbles and 9 green marbles. What’s the probability of selecting a green marble from the bag?
Total outcomes: How many marbles there are in total
9 20
What you want: How many green marbles there are
Read the question very carefully: A bag contains 6 blue marbles, 5 red marbles and 9 green marbles. What’s the probability of selecting a green or red marble from the bag?
Total outcomes: How many marbles there are in total
14 20
What you want: How many green or red marbles there are
A few questions for you: A box contains 24 different coloured balls. There are 8 red, 5 blue, 10 purple and the rest are orange. What is the probability of picking a: 𝟓 1. A blue ball? 𝟐𝟒 𝟖 𝟏 2. A red ball? 𝟐𝟒 = 𝟑 𝟐 𝟏 3. An orange ball? 𝟐𝟒 = 𝟏𝟐 4. A black ball? 0 𝟏𝟖 𝟑 5. A purple or red ball? 𝟐𝟒 = 𝟒 Will any of your answers simplify?
SUCCESS CRITERA: WHERE ARE WE NOW? Level F2 F1 E3 E3 C3 C2 B2
B1 A3
Learning outcomes: I can put probabilities on a probability scale. I can list all the outcomes from a certain event. I can calculate the probability of an event occurring, know that the probability of all events adds to 1. I can calculate with mutually exclusive probabilities. I can use both experimental and theoretical probability. I can use relative frequency to estimate probabilities. I can calculate the probability of multiple independent events. I can use a tree diagram to solve probability problems I can use a tree diagram with compound or dependent events.
Mutually exclusive Lesson Objective: Do I understand what mutually exclusive events are and can I calculate the probability of them occuring? Grade E
What are you on about? Mutually exclusive is just the fancy maths way of saying that probability of all events is 1 (100%). In other words, if you add the probabilities of each possible outcome together the answer will be 1.
How does that work? This means that if you know the probability of an event happening, you can calculate the probability of it not happening.
For example: The probability of Brighton & Hove Albion winning their next game is 0.83. What is the probability that they will not win their next game? 1 - 0.83 = 0.17
You have a go at a couple: 1. The probability that I will pass my driving test is 0.65. What is the probability of me not passing it? 0.35 2. Find the probability of me not rolling a 6 on a dice. 𝟓 𝟔
This is often asked using a table: A fairground game asks people to pick a coloured ball from a barrel. You win if you pick a black ball. Colour Probability
Blue 0.2
Red 0.35
Yellow 0.05
Green 0.25
What is the probability of winning? Answer: 0.15
Black
SUCCESS CRITERA: WHERE ARE WE NOW? Level F2 F1 E3 E3 C3 C2 B2
B1 A3
Learning outcomes: I can put probabilities on a probability scale. I can list all the outcomes from a certain event. I can calculate the probability of an event occurring, know that the probability of all events adds to 1. I can calculate with mutually exclusive probabilities. I can use both experimental and theoretical probability. I can use relative frequency to estimate probabilities. I can calculate the probability of multiple independent events. I can use a tree diagram to solve probability problems I can use a tree diagram with compound or dependent events.
Experimental and theoretical probability Lesson Objective: Can I use experimental probability and compare it with theoretical probability? Grade C
What’s the difference? Theoretical probability is what you expect to happen before you tested it. Experimental probability is what actually happens when you do it. The more experiments you do, the closer to the theoretical results you should get.
What’s it used for? Experimental probability is used to spot whether a certain outcome or outcomes are biased. If the experimental results are so far removed from the theoretical results then you can argue that the experiment is biased.
How does it work? A dice was rolled 60 times with the results shown in the table below: Score
1
2
3
4
5
6
Frequency
12
3
8
11
21
5
Is the dice biased? Justify your answer.
Answer: You would expect all numbers to be around 10. There seem to be a lot of 5s and few 2s but there haven’t been enough rolls of the dice.
SUCCESS CRITERA: WHERE ARE WE NOW? Level F2 F1 E3 E3 C3 C2 B2
B1 A3
Learning outcomes: I can put probabilities on a probability scale. I can list all the outcomes from a certain event. I can calculate the probability of an event occurring, know that the probability of all events adds to 1. I can calculate with mutually exclusive probabilities. I can use both experimental and theoretical probability. I can use relative frequency to estimate probabilities. I can calculate the probability of multiple independent events. I can use a tree diagram to solve probability problems I can use a tree diagram with compound or dependent events.
Relative frequency Lesson Objective: Can I use relative frequency to estimate results? Grade C
What is this all about? This is using previous results to find probabilities. It can also be used to predict future results.
How does it work? A dice has been rolled 120 times and the results are shown in the table below: Score
1
2
3
4
5
6
Frequency
23
15
31
21
12
18
What is the relative frequency of rolling a 3? 31 Based on these results, what is the probability of rolling 2?
𝟏𝟓 𝟏 = 𝟏𝟐𝟎 𝟖
Based on these results, what is the probability of rolling a number bigger than 4? 𝟑𝟎 = 𝟏 𝟏𝟐𝟎
𝟒
Is this dice fair? Justify your answer. Yes, because even though the results aren’t all 20, they are close enough. Each 𝟏 probability will get closer to 𝟔 with more rolls.
SUCCESS CRITERA: WHERE ARE WE NOW? Level F2 F1 E3 E3 C3 C2 B2
B1 A3
Learning outcomes: I can put probabilities on a probability scale. I can list all the outcomes from a certain event. I can calculate the probability of an event occurring, know that the probability of all events adds to 1. I can calculate with mutually exclusive probabilities. I can use both experimental and theoretical probability. I can use relative frequency to estimate probabilities. I can calculate the probability of multiple independent events. I can use a tree diagram to solve probability problems I can use a tree diagram with compound or dependent events.
Multiple events Lesson Objective: Can I calculate the probability of multiple events happening? Grade B
What does “multiple events” mean? This is dealing with more than one event at a time, as the name suggests. They can sometimes involve “sample space” diagrams, which list all possible outcomes in a table.
Here’s a common example: Rolling two dice and finding their total. There’s 6 possible outcomes on one dice, and 6 on the other.
Dice 2
Six sides on each dice
1
2
Dice 1 3 4
1
2
3
4
5
6
7
2
3
4
5
6
7
8
3
4
5
6
7
8
9
4
5
6
7
8
9
10
5
6
7
8
9
10
11
6
7
8
9
10
11
12
5
6
The scores rolled
36 possible outcomes
Dice 2
We can use that to answer questions: 1
2
Dice 1 3 4
1
2
3
4
5
6
7
2
3
4
5
6
7
8
3
4
5
6
7
8
9
4
5
6
7
8
9
10
5
6
7
8
9
10
11
6
7
8
9
10
11
12
5
6
𝟓 𝟑𝟔 𝟐 𝟏 What is the probability of scoring 11? 𝟑𝟔 = 𝟏𝟖 What score is most likely to appear? 7
1. What is the probability of scoring 8? 2. 3.
A question for you to do: I roll a dice and flip a coin at the same time. 1. How many possible outcomes are there? 2. What is the probability that I get Heads and a 3? 3. What is the probability that I get tails and an even number? Answers: 1. 𝟏𝟐
2. 3.
𝟏 𝟏𝟐
𝟑 𝟏𝟐
=
𝟏 𝟒
SUCCESS CRITERA: WHERE ARE WE NOW? Level F2 F1 E3 E3 C3 C2 B2
B1 A3
Learning outcomes: I can put probabilities on a probability scale. I can list all the outcomes from a certain event. I can calculate the probability of an event occurring, know that the probability of all events adds to 1. I can calculate with mutually exclusive probabilities. I can use both experimental and theoretical probability. I can use relative frequency to estimate probabilities. I can calculate the probability of multiple independent events. I can use a tree diagram to solve probability problems I can use a tree diagram with compound or dependent events.
Tree diagrams Lesson Objective: Can I draw and use a tree diagram? Grade B
What is a tree diagram? A tree diagram shows all the possible outcomes of two or more events happening. It shows these by using paths or branches that could be followed. In probability the word “and” means that probabilities are multiplied. The word “or” means that probabilities are added.
What do they look like? In order to start a board game a player must roll two sixes in a row. What is the probability of rolling 2 sixes? Put the probabilities on each branch
1 6 1 6
We have 2 options: 6 and “not 6”; so 2 “branches”
Answer:
Roll 2
Roll 1
5 6
𝟏 𝟑𝟔
Outcomes
Probability
6
6, 6
1 1 𝟏 × = 6 6 𝟑𝟔
not 6
6, not 6
1 5 𝟓 × = 6 6 𝟑𝟔
6
not 6, 6
5 1 𝟓 × = 6 6 𝟑𝟔
not 6, not 6
5 5 𝟐𝟓 × = 6 6 𝟑𝟔
6 5 6 1 6 not 6 5 6
not 6
You have a go at one: I have a 2 square spinners with the numbers 1, 2, 3 and 4 on. What is the probability that I get exactly one three when I spin both? Spin 2
Spin 1
1 4 3 4 1 4
3 4
Answer:
=
Probability
3
3, 3
1 1 𝟏 × = 4 4 𝟏𝟔
not 3
3, not 3
1 3 𝟑 × = 4 4 𝟏𝟔
3
not 3, 3
3 1 𝟑 × = 4 4 𝟏𝟔
not 3, not 3
3 3 𝟗 × = 4 4 𝟏𝟔
3
1 4
𝟔 𝟏𝟔
Outcomes
𝟑 𝟖
not 3 3 4
not 3
SUCCESS CRITERA: WHERE ARE WE NOW? Level F2 F1 E3 E3 C3 C2 B2
B1 A3
Learning outcomes: I can put probabilities on a probability scale. I can list all the outcomes from a certain event. I can calculate the probability of an event occurring, know that the probability of all events adds to 1. I can calculate with mutually exclusive probabilities. I can use both experimental and theoretical probability. I can use relative frequency to estimate probabilities. I can calculate the probability of multiple independent events. I can use a tree diagram to solve probability problems I can use a tree diagram with compound or dependent events.
Tree diagrams and dependent events Lesson Objective: Can I draw and use a tree diagram where the second event depends on the first? Grade A
What are “dependent” events? These are events whose probability is dependent on what has happened previously. This usually involves something not being replaced. It is imperative that you read the question carefully to check whether there is “replacement”.
A typical question: A bag contains 5 blue counters, 3 red counters and 2 green counters. If I take out one counter, without replacement, then another, what is the probability that they will be the same colour? Put on the probabilities
We have 3 options so need 3 “branches”
5 10
3 10
2 10
B
R
G
4 9 3 9 2 9
B R G
5 9 2 9 2 9 5 9 3 9
G
1 9
G
B R
B R
The probabilities of the second pick change depending upon what the first pick was. Answer: 5 4 20 Blue, Blue: 10 × 9 = 90 3
2
6
Red, Red: 10 × 9 = 90 2
1
2
Green, Green: 10 × 9 = 90 𝟐𝟖
𝟏𝟒
Total: 𝟗𝟎 = 𝟒𝟓
Don’t simplify until the end!
A typical question: A simple lottery involves a player picking 2 counters from a bag that has 20 counters numbered from 1 to 20. To win a player must pick, without replacement, 2 counters with a value over 15. What is the probability that a player only picks at least 1 counter over 15? 4 19 5 20
15 20
A
B
Answer: 5 4 20 Above, Above: 20 × 19 = 380
A
Below, Above: 20 × 19 = 380
A 15 19 5 19
B
5
15
75
15
5
75
Above, Below: 20 × 19 = 380
𝟏𝟕𝟎
14 19
B
𝟏𝟕
Total: 𝟑𝟖𝟎 = 𝟑𝟖
