Probability, Year 9: Venn diagrams and two-way tables

No — a 6 is both

HintLook for a single face that fits the two descriptions at once.

The 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.

0.8

HintA single counter cannot be two colours at once, so the two chances can be combined directly.

The 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.

4 and 6 are counted twice

HintWrite out the faces in each event and see which ones appear on the two lists.

The 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.

mutually exclusive

HintThe two events must have no outcome in common.

The 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 intersection

HintRoad junctions share the name: the place where two routes cross.

The 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".

13

HintFill in the overlap first, then ask how many footballers are left to place.

The 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.

9

HintWork out how many people are inside at least one circle, then compare with everyone surveyed.

The 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.

3/20

HintFind the region that means "both", and put it over everybody who could have been picked.

The 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.

union

HintThe word means a joining together, as in the name of the United Kingdom.

The 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.

9; 14

HintUse the total for walkers to fill one empty cell, then use the total for boys to fill the next.

The 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.

7/40

HintFind the one cell that fits the whole description, and compare it with the grand total.

The 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.

They are equal — both give 60

HintEvery person has been counted once in each direction.

The 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.