[Q] “and” rule is multiply, except when it isn’t

Question

Sorry for the kind of basic question but this is a probability question from my high school’s Stats midterm.

Given a table in which 44 people were polled about their favorite food choice, organized by male/female.

                Pasta      Pizza    Chicken
Female       4            12          7

Male           6              7          8


The question posed is P(Pizza and Male). So, I found the probability of liking pizza (19/44) and the probability of being male (21/44) and applied the “and” rule, multiplying them together to get 399/1936.

The teacher said that’s wrong and that using the table we can see that the probability is 7/44 since 7 people are male and like pizza. The two numbers are not even close to each other.

So which is the correct way to do this problem? Is the “and” rule only used when we have an incomplete data set? Does it give us a “standard” probability, or an extrapolated one?

Answer 1

Probabilities for intersections multiply like you would expect,

    P(A and B) = P(A)P(B),

**if and only if A and B are independent events**.

So if you know that two events are independent, like a coin flip coming up heads and a die rolling 1, then you can find the probability of both events happening by multiplying. But events are not independent in general, and if you're given a table like this, you should look at the count in the intersection of Male and Pizza instead of multiplying probabilities. In fact, even if "Male" and "Pizza" were independent, you would get the same answer both ways and looking up a number in the table is easier than multiplying fractions, so you should do that anyway :)

Answer 2

Fixing up the table so it's readable:

|       | Pasta   |   Pizza  |  Chicken
|-|-:|-:|-:|
|Female |     4  |  12   |  7
| Male   |  6  |   7  |   8

(see at the end for how to achieve this)

...

The multiplication rule you're using (i.e. P(A and B) = P(A) x P(B) ) is for *independent* events only. That's not the case here.

---

    |       | Pasta   |   Pizza  |  Chicken
    |-|-:|-:|-:|
    |Female |     4  |  12   |  7
    | Male   |  6  |   7  |   8

Answer 3

Others have already explained why, but with a table and question like that, you should read the table.  You look at where the Male and Pizza rows and columns intersect and divide by the total number of people.

Answer 4

If you really want to do the probability multiplication method rather than readout of table directly, make sure you factor our your joint probability properly.

P(Pizza, Male) = P(Male) * P(Pizza | Male)
or
P(Pizza, Male) = P(Pizza) * P(Male | Pizza)

And then do the calculation reading out values from the table.

However, you can directly readout it from the table so the factorisation method is not needed here.