Two Way Tables (Cambridge O Level Maths)

What are two-way tables?

• Two-way tables allow us to consider two characteristics within a set of data
  • For example, we may be interested in the number of students studying Spanish or German
    • We may also be interested in how many of those students are in year 12 and how many are in year 13
  • Spanish/German would be one characteristic in the two-way table, year 12/13 would be the second
• One of the characteristics will be represented by the columns, the other by the rows
• A two-way table should include row totals and column totals
  • The row/column totals are sometimes called marginal (or sub-) totals
    
  • Where the row totals and column totals meet, we have the grand total
  • Marginal totals can be really useful in two-way table questions
    • If they're not mentioned, or not included in a given table, add them in!

• Once a two-way table is completed, with marginal totals, the values within it can be used to determine probabilities

How do I draw and complete a two-way table?

• To construct a two-way table from information given in words in a question
  • identify the two characteristics
  • use rows for one characteristic and columns for the other
  • add an extra row and column for the marginal totals (and grand total)
• Work your way through each sentence in the question
  • fill in any values you can directly from the information given
  • be prepared to come back to any information that cannot be put into the two-way table directly
    • some information may need combining in order to deduce a value

How do I find probabilities from a two-way table?

• This is a matter of going from the words used in the question to probability phrases
  • Aim to rephrase the question in your head using AND and/or OR statements
  • e.g. The probability of selecting a year 12 student who studies German is P("year 12 AND German")
• Once you are clear what parts of the two-way table are required you can begin to write down the probability
  
  • the numerator will be taken from the main body of the two-way table
    • for "year 12 AND German" this would be the cell in the table where the row/column for "year 12" meets the row/column for "German"
  • the denominator will be the total of the group we are choosing from
    • this could be either a row/column (marginal) total or the grand total 
    • if we are choosing from the whole group it would be the grand total
    • if we are choosing from just year 12 students, say, it would be the total for the year 12 row/column

How do I work with two-way tables and conditional probability?

• Probabilities that only involve a subset of the things in a two-way table are called conditional probabilities
  • For example with male and female students studying various languages, you might want to know the probability that a male student studies German
  • This would require considering only the 'male' row or column of the two-way table when finding the probability
• Conditional probability questions are often (but not always!) introduced by the expression 'given that...'
  • For example 'Find the probability that a randomly chosen student studies German, given that the student is male'
  • The answer would be the number of 'male AND German' students divided by the total for the male row or column
• Conditional probabilities are sometimes written using the 'straight bar' notation 
  • That is read as 'the probability of A given B'
  • For example  would be the probability that a student studies German, given that the student is male