Every card in Probability, Year 9: Venn diagrams and two-way tables
The whole deck, in order — so you can read it through before your child ever sees it.
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- A fair six-sided dice is rolled once. Are the events "an even number" and "a number greater than 4" mutually exclusive?
No — a 6 is both
HintLook for a single face that fits the two descriptions at once.
WhyEven numbers are 2, 4 and 6; numbers greater than 4 are 5 and 6. The two lists share the 6, so one roll can make both events happen, and events that can happen together are not mutually exclusive.
- A bag holds only red, blue and green counters. One counter is picked at random. The probability that it is red is 0.3 and the probability that it is blue is 0.5. What is the probability that it is red or blue? Give a decimal.
0.8
HintA single counter cannot be two colours at once, so the two chances can be combined directly.
WhyRed and blue cannot both happen on one pick, so the probabilities are added: 0.3 + 0.5. That leaves 0.2 for green, because the three colours cover every possibility.
- A fair six-sided dice is rolled once. P(even) = 3/6 and P(greater than 3) = 3/6. Why is P(even or greater than 3) not 3/6 + 3/6 = 1?
4 and 6 are counted twice
HintWrite out the faces in each event and see which ones appear on the two lists.
WhyEven gives 2, 4, 6 and greater than 3 gives 4, 5, 6. Together they cover 2, 4, 5 and 6, which is four faces, so the probability is 4/6. Adding the two fractions counts the shared faces once too often.
- You may find P(A or B) by adding P(A) and P(B) only when events A and B are ____.
mutually exclusive
HintThe two events must have no outcome in common.
WhyWhen two events share no outcomes, nothing is counted twice, so their probabilities can simply be added. When they overlap, count the outcomes directly so that each one is counted once.
- The region of a Venn diagram where two circles overlap, holding the items that belong to both sets
The intersection
HintRoad junctions share the name: the place where two routes cross.
WhyAnything written in the overlap is in set A and in set B at once. In a probability question it is the region that answers "both".
- In a class of 30 pupils, 17 play football, 10 play tennis and 4 play both. A Venn diagram has one circle for football and one for tennis. How many pupils go in the part of the football circle that is outside the tennis circle? (number only)
13
HintFill in the overlap first, then ask how many footballers are left to place.
WhyThe 4 who play both go in the overlap. They are already among the 17 footballers, so the football-only region holds 17 − 4. Always fill a Venn diagram from the middle outwards.
- Of 40 people surveyed, 23 own a dog, 14 own a cat and 6 own both. How many own neither a dog nor a cat? (number only)
9
HintWork out how many people are inside at least one circle, then compare with everyone surveyed.
WhyDog only is 23 − 6 = 17 and cat only is 14 − 6 = 8. With the 6 in the overlap that makes 31 people inside the circles, so 40 − 31 are outside both.
- A Venn diagram sorts the 20 pupils in a class. 8 are in the 'has a brother' circle only, 5 are in the 'has a sister' circle only, 3 are in the overlap and 4 are outside both circles. One pupil is picked at random. What is the probability that the pupil has both a brother and a sister? Give a fraction in its simplest form.
3/20
HintFind the region that means "both", and put it over everybody who could have been picked.
WhyThe overlap holds the pupils with a brother and a sister. Any of the 20 pupils could be picked, including the 4 outside the circles, so the total is the whole class.
- In a Venn diagram, everything that lies inside at least one of the two circles makes up the ____ of the two sets.
union
HintThe word means a joining together, as in the name of the United Kingdom.
WhyThe union is everything in A or B or both: the two circles taken as one region. The items left outside it, inside the rectangle, are in neither set.
- A two-way table sorts 50 pupils by whether they are girls or boys and by whether they walk to school. There are 27 girls, of whom 12 walk, and 23 boys. Altogether 21 pupils walk. So ____ boys walk and ____ boys do not walk.
9; 14
HintUse the total for walkers to fill one empty cell, then use the total for boys to fill the next.
WhyEvery row and column of a two-way table must add up to its total. The walkers total 21 and 12 of them are girls, which fixes the boys who walk; the 23 boys then fix the boys who do not.
- A two-way table sorts the 40 pupils in a year group. 10 boys and 14 girls have a school lunch; 9 boys and 7 girls bring a packed lunch. One of the 40 pupils is picked at random. What is the probability of picking a girl who brings a packed lunch? Give a fraction in its simplest form.
7/40
HintFind the one cell that fits the whole description, and compare it with the grand total.
WhyThe cell where the "girl" row meets the "packed lunch" column holds 7 pupils. The pupil is picked from the whole year group, so the 7 is put over the grand total of 40.
- A two-way table sorts 60 people by age group (rows) and by their answer to a question (columns). You add up the row totals, and separately add up the column totals. What should be true of the two results?
They are equal — both give 60
HintEvery person has been counted once in each direction.
WhyThe rows split the same people one way and the columns split them another, so each set of totals must add back to everyone in the table. If the two disagree, a cell has been filled in wrongly.
1★ KS3-MATH-PRB-0013
2★ KS3-MATH-PRB-0014
3★ KS3-MATH-PRB-0015
4★ KS3-MATH-PRB-0016
5★ KS3-MATH-PRB-0017
6★ KS3-MATH-PRB-0018
7★ KS3-MATH-PRB-0019
8★ KS3-MATH-PRB-0020
9★ KS3-MATH-PRB-0021
10★ KS3-MATH-PRB-0022
11★ KS3-MATH-PRB-0023
12★ KS3-MATH-PRB-0024
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