Year 10 Standard · Unit 4 Sets, Venn diagrams and probability · Lesson 5 of 6

Which group are you counting out of?

A two-way table and a Venn diagram hold the same counts. This lesson reads probabilities from both, and asks what changes when two events cannot happen together.

Statistics and ProbabilityNo calculator

Warm-up

Three quick ones from earlier work. Try each in your head or on paper before you open the answer.

  1. From Lesson 4: 3/5 × 2/3 = ?
    Answer2/5. Multiply the tops (6) and the bottoms (15), then divide both by 3.
  2. From Lesson 2: n(A) = 14, n(B) = 9 and n(A ∩ B) = 4. What is n(A ∪ B)?
    Answer19. 14 + 9 − 4 = 19.
  3. From Year 7: write 18/24 as a percentage.
    Answer75%. 18/24 = 3/4.
Words used in this lesson
two-way table
a table that classifies the same group by two features at once. For example: pupils by year group and how they travel.
within a group
counting only the people or outcomes in one named group. For example: chess players among the females.
mutually exclusive
two events that cannot both happen at the same time. For example: a coin landing heads and landing tails on one toss.
union
outcomes in A or in B or in both, written A ∪ B. For example: a card that is odd or greater than 15.
intersection
outcomes in both A and B, written A ∩ B. For example: a card that is a square and also odd.
total
the number in a whole row, column or group. For example: all 120 pupils.

Two coaches, one table

A coach in Nairobi looks at the school chess club and says: “Most of the people who play chess here are girls.” Her colleague looks at the same school and says: “Most of the girls here do not play chess.”

Predict first: can both of them be right? What would you need to know about the school to decide? Use the first figure, which holds one table of 80 people, to test it.

chess
29in the event
80in the group counted out of

P(chess) = 29/80: 29 of all 80 people play chess.

Show me the steps
  1. P(chess) = 29/80: the whole group is all 80 people, and 29 of them play chess.
  2. Slide to P(female and chess): only the cell in both, 20, out of the whole 80.
  3. Slide to P(female or chess): three cells, 20 + 25 + 9 = 54, out of 80. Check: 45 + 29 − 20 = 54.
  4. Slide to P(chess, among females): the group is now only the 45 females, so 20 of 45, not 20 of 80.
  5. Slide to P(female, among chess players): the group is now the 29 chess players, so 20 of 29. Most chess players are female, but fewer than half the females play chess.

Say which group you are counting out of.

  • A two-way table and a Venn diagram hold the same counts. P(A) is a count divided by the total.
  • Within a group: if you are told the person is in group B, only B counts. The group's total is the new bottom number.
  • P(A ∪ B) = P(A) + P(B) − P(A ∩ B). Subtract the overlap once, because adding counted it twice.
  • Events are mutually exclusive if they cannot happen together: P(A ∩ B) = 0, so P(A ∪ B) = P(A) + P(B).
7
28/50P(A or B)
nomutually exclusive?

P(A) + P(B) = 20/50 + 15/50 = 35/50. Take away the 7/50 counted twice: P(A or B) = 28/50.

Show me the steps
  1. 20 in A, 15 in B and 7 in both: P(A or B) = (20 + 15 − 7)/50 = 28/50.
  2. Slide the overlap to 0: the circles do not touch. A and B cannot happen together, so they are mutually exclusive, and P(A or B) = 20/50 + 15/50 = 35/50.
  3. Slide the overlap to 10: P(A or B) = (35 − 10)/50 = 25/50. The bigger the overlap, the smaller the union.
  4. Slide the overlap to 3: P(A or B) = (35 − 3)/50 = 32/50. Mutually exclusive is a special case of the rule where the subtracted part is 0.

“Of the girls” and “of the chess players” are different fractions. The top number can be the same, 20, while the bottom number is 45 in one and 29 in the other. Check which group the sentence is about before choosing a bottom number. And mutually exclusive does not mean “unrelated”: events that cannot happen together are very much related, because one happening rules the other out.

Worked example: reading a table

In a survey of 120 pupils in Quito, 70 are in Year 10 and 50 are in Year 11. 45 of the Year 10 pupils and 20 of the Year 11 pupils come to school by bus. Find P(bus), P(Year 11 and bus) and, among Year 11 pupils, the probability of a bus.

  1. Total on the bus: 45 + 20 = 65, so P(bus) = 65/120 = 13/24.
  2. In Year 11 and on the bus: 20 pupils out of all 120, so P = 20/120 = 1/6.
  3. Among Year 11 only: the group is the 50 Year 11 pupils, and 20 take the bus, so 20/50 = 2/5.

Your go. The first step is given. Using the same survey, find P(Year 10 or bus).

  1. 70 are in Year 10, 65 take the bus, and 45 are in both.
  2. Use P(A or B) = P(A) + P(B) − P(A and B), or count the cells.
Check the answer

70 + 65 − 45 = 90, so P = 90/120 = 3/4. A common slip is 135/120, which adds without taking the overlap away and gives a probability above 1.

Check before you go on

1. A card is chosen at random from cards numbered 1 to 20. Which pair of events is mutually exclusive?

Your turn

Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. In a survey of 100 people in Hanoi, 31 are members of a bike-share scheme and rode last week, 19 are members who did not ride, 17 are not members and rode, and 33 are not members and did not ride. A person is chosen at random. What is the probability, as a percentage, that they did not ride last week?

3. Using the same survey of 100 people, a member of the scheme is chosen at random. What is the probability, as a percentage, that this member rode last week?

Level 2 · Building

4. A person is chosen at random. P(likes tea) = 0.55, P(likes coffee) = 0.4 and P(likes both) = 0.2. What is the probability that the person likes neither?

5. In a school of 150 pupils, 90 are in the swim club, 60 are in the chess club and 36 are in both. A chess club member is chosen at random. What is the probability, as a percentage, that this member is also in the swim club?

6. For events A and B, P(A) = 0.4, P(B) = 0.35 and P(A ∪ B) = 0.75. Are A and B mutually exclusive?

Level 3 · Stretch

7. P(A) = 0.5, P(B) = 0.3 and P(A ∩ B) = 0.1. What is P(A ∪ B′)?

8. A spinner lands on red, blue or green. P(red) = 2x, P(blue) = x + 0.1 and P(green) = 0.4 − x. What is x?

Something to think about

A coach says: “Most of the members of our chess club are girls, so most girls play chess.” Describe a school where the first part is true and the second part is false.

One way to think about it

Take 200 girls and 100 boys in a school, with a chess club of 30 members: 20 girls and 10 boys. Most of the members (20 of 30) are girls, but only 20 of the 200 girls play chess. The first claim counts out of the club, and the second counts out of all the girls, so the two fractions have different bottom numbers.

Exit ticket

9. In a music school, P(plays piano) = 0.3, P(plays violin) = 0.2 and P(plays both) = 0.05. What is P(plays piano or violin)?