Steply

Venn and tree diagrams

Probability · 20 problems, easy to harder

▶ Watch the first one solved

How we do it

  1. Start in the middle
  2. Fill the rest
  3. Find the part we want
  4. Write the fraction
  5. Check

The steps for 40 students: 22 play soccer, 15 play tennis, 8 play both, P(both). Every solution names its steps, so you always know what comes next.

Worked examples

40 students: 22 play soccer, 15 play tennis, 8 play both, P(both) Easy

The question

Question: 40 students were asked.

22 play soccer, 15 play tennis, and 8 play both.

One is picked at random.

What is the probability they play both?

1 Start in the middle

8 play both.

They go in the middle, where the circles overlap.

2 Fill the rest

22 play soccer, but 8 of them are already in the middle.

22 minus 8 is 14.

soccer only = 22 − 8 = 14

Tennis only: 15 minus 8 is 7.

tennis only = 15 − 8 = 7

Neither: everyone else.

40 minus 14, 8 and 7 is 11.

neither = 40 − 14 − 8 − 7 = 11

3 Find the part we want

Both: the middle, 8.

4 Write the fraction

So the probability is 8/40.

P(both) = 840

Both 8 and 40 can be divided by 8.

That makes it simpler.

P(both) = 8 ÷ 840 ÷ 8

So it is 1/5.

P(both) = 15

5 Check

Let's check: the four parts add up to 14 plus 8 plus 7 plus 11.

That is 40, everyone.

So the answer is right!

14 + 8 + 7 + 11 = 40

Let's look back.

First: start in the middle.

Then: fill the rest.

Then: find the part we want.

Then: write the fraction.

Last: check.

Answer
P(both) = 15

▶ Watch it solved

40 students: 22 play soccer, 15 play tennis, 8 play both, P(neither) Medium

The question

Question: 40 students were asked.

22 play soccer, 15 play tennis, and 8 play both.

One is picked at random.

What is the probability they play neither?

1 Start in the middle

8 play both.

They go in the middle, where the circles overlap.

2 Fill the rest

22 play soccer, but 8 of them are already in the middle.

22 minus 8 is 14.

soccer only = 22 − 8 = 14

Tennis only: 15 minus 8 is 7.

tennis only = 15 − 8 = 7

Neither: everyone else.

40 minus 14, 8 and 7 is 11.

neither = 40 − 14 − 8 − 7 = 11

Careful!

That takes away the 8 who play both twice.

Start in the middle.

Careful: not 40 − 22 − 15 ✗

3 Find the part we want

Neither: outside both circles, 11.

4 Write the fraction

So the probability is 11/40.

P(neither) = 1140

5 Check

Let's check: the four parts add up to 14 plus 8 plus 7 plus 11.

That is 40, everyone.

So the answer is right!

14 + 8 + 7 + 11 = 40

Let's look back.

First: start in the middle.

Then: fill the rest.

Then: find the part we want.

Then: write the fraction.

Last: check.

Answer
P(neither) = 1140

▶ Watch it solved

40 students: 22 play soccer, 15 play tennis, 8 play both, P(tennis… Harder

The question

Question: 40 students were asked.

22 play soccer, 15 play tennis, and 8 play both.

One is picked at random.

What is the probability they play tennis, if they play soccer?

1 Start in the middle

8 play both.

They go in the middle, where the circles overlap.

2 Fill the rest

22 play soccer, but 8 of them are already in the middle.

22 minus 8 is 14.

soccer only = 22 − 8 = 14

Tennis only: 15 minus 8 is 7.

tennis only = 15 − 8 = 7

Neither: everyone else.

40 minus 14, 8 and 7 is 11.

neither = 40 − 14 − 8 − 7 = 11

3 Find the part we want

Given they play soccer: only the soccer circle counts.

22 in it.

Of those, 8 play tennis too.

The soccer circle: 14 plus 8 is 22.

That is the new bottom.

all soccer = 14 + 8 = 22

Careful!

Given soccer: the bottom is only the 22 in the soccer circle.

Not all 40.

Careful: not 840 ✗

4 Write the fraction

So the probability is 8/22.

P(tennis | soccer) = 822

Both 8 and 22 can be divided by 2.

That makes it simpler.

P(tennis | soccer) = 8 ÷ 222 ÷ 2

So it is 4/11.

P(tennis | soccer) = 411

5 Check

Let's check: the four parts add up to 14 plus 8 plus 7 plus 11.

That is 40, everyone.

So the answer is right!

14 + 8 + 7 + 11 = 40

Let's look back.

First: start in the middle.

Then: fill the rest.

Then: find the part we want.

Then: write the fraction.

Last: check.

Answer
P(tennis | soccer) = 411

▶ Watch it solved

Practice

Try each one, then open the answer. Stuck? Open the problem to watch every step.

Easy

  1. 40 students: 22 play soccer, 15 play tennis, 8 play both, P(both)
    Answer
    P(both) = 15
  2. 30 people: 18 like tea, 12 like coffee, 5 like both, P(tea only)
    Answer
    P(tea only) = 1330
  3. 50 students: 28 take art, 20 take music, 10 take both, P(neither)
    Answer
    P(neither) = 625
  4. 25 people: 12 own cats, 10 own dogs, 4 own both, P(dogs only)
    Answer
    P(dogs only) = 625
  5. P(rain) = 0.3, P(late given rain) = 0.4, P(late given no rain) = 0.1, find P(late)
    Answer
    P(late) = 0.19
  6. P(sun) = 0.6, P(walk given sun) = 0.8, P(walk given no sun) = 0.3, find P(walk)
    Answer
    P(walk) = 0.6
  7. P(test) = 0.5, P(pass given test) = 0.9, P(pass given no test) = 0.6, find P(pass)
    Answer
    P(pass) = 0.75

Medium

  1. 40 students: 22 play soccer, 15 play tennis, 8 play both, P(neither)
    Answer
    P(neither) = 1140
  2. 30 people: 18 like tea, 12 like coffee, 5 like both, P(tea or coffee)
    Answer
    P(tea or coffee) = 56
  3. 60 students: 35 read comics, 25 read novels, 12 read both, P(not novels)
    Answer
    P(not novels) = 712
  4. 36 people: 20 like pizza, 18 like pasta, 9 like both, P(pizza only)
    Answer
    P(pizza only) = 1136
  5. P(snow) = 0.2, P(late given snow) = 0.5, P(late given no snow) = 0.1, find P(late)
    Answer
    P(late) = 0.18
  6. P(win) = 0.4, P(party given win) = 0.9, P(party given no win) = 0.2, find P(party)
    Answer
    P(party) = 0.48
  7. P(bus) = 0.7, P(early given bus) = 0.6, P(early given no bus) = 0.3, find P(early)
    Answer
    P(early) = 0.51

Harder

  1. 40 students: 22 play soccer, 15 play tennis, 8 play both, P(tennis given soccer)
    Answer
    P(tennis | soccer) = 411
  2. 50 students: 28 take art, 20 take music, 10 take both, P(art given music)
    Answer
    P(art | music) = 12
  3. 30 people: 18 like tea, 12 like coffee, 5 like both, P(coffee given tea)
    Answer
    P(coffee | tea) = 518
  4. P(rain) = 0.3, P(late given rain) = 0.4, P(late given no rain) = 0.1, find P(rain given late)
    Answer
    P(rain | late) ≈ 0.632
  5. P(sun) = 0.6, P(walk given sun) = 0.8, P(walk given no sun) = 0.3, find P(sun given walk)
    Answer
    P(sun | walk) = 0.8
  6. P(snow) = 0.2, P(late given snow) = 0.5, P(late given no snow) = 0.1, find P(snow given late)
    Answer
    P(snow | late) ≈ 0.556