Probability from two-way tables Problem set A Worked solutions

The table shows the tickets sold at a museum in one week. One visitor is chosen at random. Find the probability that:

1. (a) the visitor bought a senior ticket;

   • Senior row: 18 + 28 = 46
   • Grand total: 75 + 79 + 46 = 200
   • \(P=\frac{46}{200}=\frac{23}{100}=0.23\)

2. (b) the visitor came on a weekday.

   • Weekday column: 34 + 52 + 18 = 104
   • Grand total: 200
   • \(P=\frac{104}{200}=\frac{13}{25}=0.52\)

The table shows the phone type and year group of the students in Years 7 to 9 who answered a survey. One student is chosen at random. Find the probability that the student:

1. (a) has an iPhone and is in Year 8;

   • Both properties is one cell: iPhone and Year 8 = 36
   • Grand total: 180
   • \(P=\frac{36}{180}=\frac{1}{5}=0.2\)

2. (b) has an Android phone or is in Year 9 (or both).

   • Android row total 75, Year 9 column total 59; the cell they share (19) is counted twice
   • Number: 75 + 59 − 19 = 115
   • \(P=\frac{115}{180}=\frac{23}{36}\)

The table shows the favourite sport and the gender of the members of a youth club.

1. (a) One member who has tennis as a favourite is chosen at random. Find the probability that this member is a girl.

   • Only the Tennis row is left: total 25 + 15 = 40
   • Of these, Girls: 25
   • \(P=\frac{25}{40}=\frac{5}{8}=0.625\)

2. (b) One member who is a boy is chosen at random. Find the probability that this member has swimming as a favourite.

   • Only the Boys column is left: total 42 + 15 + 28 = 85
   • Of these, Swimming: 28
   • \(P=\frac{28}{85}\)

The table shows the colour and size of the shirts in a shop. One shirt that is medium is chosen at random. Find the probability that this shirt is blue.

• Only the Medium column is left: 30 + 20 = 50
• Of these, Blue: 30
• \(P=\frac{30}{50}=\frac{3}{5}=0.6\)

The table shows the clubs joined by the students of a sixth form. There are 120 students in all, and the number in the Drama row, Year 12 column is k.

1. (a) Find the value of k.

   • Known cells total 100, so k = 120 − 100 = 20

2. (b) One student is chosen at random. Find the probability that the student joined drama.

   • Drama row: 20 + 25 = 45
   • \(P=\frac{45}{120}=\frac{3}{8}=0.375\)

3. (c) One student who is in Year 13 is chosen at random. Find the probability that this student joined robotics.

   • Year 13 column total: 60
   • Of these, Robotics: 17
   • \(P=\frac{17}{60}\)

The table shows how the staff and students of a college travel in. One person is chosen at random.

1. (a) Find the probability that the person does not cycle.

   • Cycle row total: 40; so the others: 150 − 40 = 110
   • \(P=\frac{110}{150}=\frac{11}{15}\)

2. (b) Given that the person does not cycle, find the probability that the person is a student.

   • Keep the other rows (Bus, Walk): total 110
   • Of these, Student: 43 + 25 = 68
   • \(P=\frac{68}{110}=\frac{34}{55}\)

One hundred and twenty students sat a driving theory test. The table is partly filled in.

1. (a) Copy the table and find the missing numbers.

   • Passed row: 78 − 65 = 13
   • Revised column: 90 − 65 = 25
   • Failed row: 42 − 25 = 17

2. (b) One student is chosen at random. Find the probability that the student passed and did not revise.

   • One cell: 13; grand total 120
   • \(P=\frac{13}{120}\)

3. (c) One student who revised is chosen at random. Find the probability that this student failed.

   • Revised column total: 90
   • Of these, Failed: 25
   • \(P=\frac{25}{90}=\frac{5}{18}\)