Probability flashcards
13 cards on Probability: the key points, worked examples and the slips to watch. Swipe through them, or print them.
Card 1 of 13
Probability
The probability scale
What does the probability scale run from, and how do you find the probability of something not happening?
Answer it in your head, then turn the card over.
The probability scale
Key points
- Every probability is from 0 (impossible) to 1 (certain); is an even chance.
- The probabilities of all the possible outcomes add up to 1.
- So P(not A) = 1 − P(A), and a missing probability is 1 minus the ones you know.
- Write a probability as a fraction, a decimal or a percentage.
Example
P(rain) = 0.3, so P(no rain) = 1 − 0.3 = 0.7
A spinner is red, blue, green or yellow: red 0.2, blue 0.35, green 0.15
P(yellow) = 1 − (0.2 + 0.35 + 0.15) = 1 − 0.7 = 0.3
Watch out
Never write a probability as a ratio (3 : 8) or in words (3 in 8): write .
Probability
Listing outcomes
How do you list all the outcomes of two events without missing any?
Answer it in your head, then turn the card over.
Listing outcomes
Key points
- Work in a fixed order: fix the first item and run through every option for the second.
- Then move the first item on and repeat.
- The number of outcomes is the number for the first event × the number for the second.
A coin and a 1 to 3 spinner
H1, H2, H3, T1, T2, T3
2 × 3 = 6 outcomes
Watch out
HT and TH are different outcomes: two coins have 4 outcomes (HH, HT, TH, TT), not 3.
Probability
Sample space diagrams
How do you use a sample space diagram to find a probability for two dice?
Answer it in your head, then turn the card over.
Sample space diagrams
Key points
- Draw a grid: one dice along the top, the other down the side.
- Fill each cell with the result, such as the total: 6 × 6 = 36 cells.
- Count the cells you want, then divide by the number of cells.
P(total of 7) with two dice
7 = 1 + 6, 2 + 5, 3 + 4, and each the other way round
6 cells out of 36: =
Watch out
The 11 totals from 2 to 12 are not equally likely, so P(total of 7) is not .
Probability
Equally likely outcomes
How do you find a probability when every outcome is equally likely?
Answer it in your head, then turn the card over.
Equally likely outcomes
Key points
- Probability = number of outcomes you want ÷ total number of outcomes.
- It only works when every outcome has the same chance, like a fair dice or a counter picked at random.
- A biased spinner needs its probabilities given, or estimated from an experiment.
Example
A prime on a fair dice: 2, 3 and 5, so =
A bag of 4 red, 5 blue and 3 green: P(blue) =
Watch out
1 is not a prime number: counting it gives for a prime on a dice, not .
Probability
Relative frequency
How do you estimate a probability from the results of an experiment?
Answer it in your head, then turn the card over.
Relative frequency
Key points
- Relative frequency = number of times it happened ÷ number of trials.
- It is an estimate of the probability, used when outcomes cannot be counted as equally likely.
- More trials give a more reliable estimate, because runs of luck even out.
13 heads in 40 flips
Relative frequency of heads = = 0.325
Watch out
Divide by all the trials (40 flips), not by the number of times it did not happen (27 tails).
Probability
Expected frequency
How many times would you expect an outcome to happen in a number of trials?
Answer it in your head, then turn the card over.
Expected frequency
Key points
- Expected frequency = probability × number of trials.
- It starts from a probability and predicts a count: relative frequency goes the other way.
- It is an average: the real count will usually be a little different.
Example
P(win) = 0.3, 200 games: 0.3 × 200 = 60 wins
Sixes in 120 rolls of a fair dice: × 120 = 20
Watch out
Multiply the probability by the trials: 200 ÷ 0.3 gives 667, more wins than games.
Probability
Two-way tables
How do you complete a two-way table and use it to find a probability?
Answer it in your head, then turn the card over.
Two-way tables
Key points
- Fill in what you are given, then use the totals: a gap is a total minus the rest of its row or column.
- Each gap you fill can unlock another; check the grand total at the end.
- For a probability, divide by the total the question is about.
60 students: 25 girls, 40 walk to school
Boys = 60 − 25 = 35
15 girls walk, so boys who walk = 40 − 15 = 25
Boys who do not walk = 35 − 25 = 10
P(a boy who walks) = =
Watch out
Read which total to divide by: the whole table, one row or one column.
Probability
Frequency trees
What goes on a frequency tree, and how do you complete one?
Answer it in your head, then turn the card over.
Frequency trees
Key points
- A frequency tree holds the numbers of people or things, not probabilities.
- At each split, the branches add up to the number before it.
- Turn a branch into a probability by dividing it by the right total.
200 people: 120 adults, 45 with a pet
Children = 200 − 120 = 80
Adults with no pet = 120 − 45 = 75
50 children have a pet, so 80 − 50 = 30 do not
P(a child with a pet) = =
Watch out
Check every split adds back up: 45 + 75 must make the 120 adults.
Probability
Venn diagrams
How do you fill in a Venn diagram from the totals you are given?
Answer it in your head, then turn the card over.
Venn diagrams
Key points
- Put the overlap in first: the number in both groups.
- Subtract it from each group's total to find the parts in one group only.
- Whatever is left of the grand total goes outside both circles.
Of 30: 18 football, 12 tennis, 5 both
Football only = 18 − 5 = 13
Tennis only = 12 − 5 = 7
Neither = 30 − 13 − 7 − 5 = 5
P(football only) =
Watch out
Writing 18 outside the overlap counts the 5 who play both twice.
Probability
Tree diagrams: one event and then another
How do you use a tree diagram to find the probability of two things both happening?
Answer it in your head, then turn the card over.
Tree diagrams: one event and then another
Key points
- Each set of branches shows one event; its probabilities add up to 1.
- Multiply along the path for one event and then another.
- For independent events, the second set of branches is the same on every path.
Two fair dice
P(six and six) = × =
P(no sixes) = × =
Watch out
Never add for "and": + = would make two sixes more likely than one.
Probability
Tree diagrams: adding paths
How do you find a probability on a tree diagram when more than one path fits?
Answer it in your head, then turn the card over.
Tree diagrams: adding paths
Key points
- Multiply along each path that fits, then add those paths together.
- "One of each" usually has two paths: red then blue, and blue then red.
- For "at least one", 1 − P(none) is often quickest.
- Check: the probabilities at the ends of all the paths add up to 1.
P(red) = each time
One of each: × + ×
= + =
At least one red: 1 − × =
Watch out
Finding only one path for "one of each" gives , half the real answer.
Probability
Picking without replacement
How do the probabilities change when an item is taken and not put back?
Answer it in your head, then turn the card over.
Picking without replacement
Key points
- Events are dependent when the first outcome changes the probability of the second.
- Without replacement there is one fewer item, so the bottom of each second-branch fraction goes down by 1.
- The top goes down by 1 too when the same colour is picked again.
- With replacement, the bag is as it started: the events are independent.
5 red and 3 blue: P(two reds)
× =
=
Watch out
Keeping 8 on the bottom, × = , treats the counter as put back.
Probability
HigherConditional probability
How do you answer a "given that" probability question?
Answer it in your head, then turn the card over.
Conditional probability
Key points
- "Given that" narrows the whole down to the group you are told about.
- Divide by that group's total, not the whole total.
- On a tree diagram, the second set of branches are conditional probabilities.
Formula
P(B given A) = P(A and B) ÷ P(A)
Example
30 students: 12 play football, 3 of those play tennis
P(tennis given football) = =
5 red, 3 blue, red first: P(second red) =
Watch out
Dividing by all 30 students gives = , which answers a different question.
Print these cards
Four cards to an A4 sheet, each the size of a standard A6 revision card.
One-sided
Each card prints on one face: the question at the top, the notes under it.
Two-sided
Questions on the front, notes on the back. Print double-sided and flip on the long edge, then cut along the lines.
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