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Chapter 13: Forming Associations with Two-Way Tables 227

for the personal cellphone use variable. Everyone has to be classified
somewhere.

Be advised that some probabilities aren’t useful in terms of discovering infor-
mation about the population in general. For example, P(F) = 0.50 because the
researchers determined ahead of time that they wanted exactly 508 females
and exactly 508 males. The fact that 50 percent of the sample is female and
50 percent of the sample is male doesn’t mean that in the entire population of
cellphone users 50 percent are males and 50 percent are females. If you want
to study what proportion of cellphone users are females and males, you need
to take a combined sample instead of two separate ones and see how many
males and females appear in the combined sample.

Joint probabilities

A joint probability gives the probability of the intersection of two categories,
one from the row variable and one from the column variable. It’s the probabil-
ity that someone selected from the whole group has two particular character-
istics at the same time. In other words, both characteristics happen jointly,
or together. You find a joint probability by taking the cell count for those
having both characteristics and dividing by the grand total.

Here are the four joint probabilities in the cellphone example:

✓ The probability that someone from the entire group is male and uses his
cellphone for personal calls is , meaning that 32 percent of all
the cellphone users in this sample are males using their cellphones for
personal calls.
✓ The probability that someone from the entire group is male and doesn’t
use his cellphone for personal calls is .

✓ The probability that someone from the entire group is female and makes
personal calls with her cellphone is .
✓ The probability that someone from the entire group is female and
doesn’t make personal calls with her cellphone is .

The notation for the joint probabilities listed is as follows, where + repre-
sents the intersection of the two categories listed:

✓ P(M + Yes) = 0.32
✓ P(M + No) = 0.18
✓ P(F + Yes) = 0.42
✓ P(F + No) = 0.08

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