# Consider a two-way table that summarizes age and ice cream preference. Of the children, 7 prefer vanilla, 12 prefer chocolate, and 3 prefer butter pecan. Of the adults, 10 prefer vanilla, 31 prefer chocolate, and 8 prefer butter pecan. Are the events "chocolate" and "child" independent? Why or why not?

May 16, 2015

One way to test if an event is independent is to check if the one event given the other, causes a different result.

If we are to calculate, we have in total
$22$ Children
$49$ Adults

$71$ in total

we know that $12$ out of the $22$ children prefer chocolate.
so the probability of getting chocolate given that you are serving a child is $\frac{12}{22}$

now if we add up both the chocolate people for adults and children, we get $12 + 31 = 43$
so $43$ out of the total amount of customers prefer chocolate.
therefore, the probability of selling a chocolate ice cream is $\frac{43}{71}$

Now we can observe that $\frac{12}{22} \ne \frac{43}{71}$

which means that if you have a child coming to buy ice cream, it does have an effect on weather you will sell a chocolate ice cream.

thus we can say that the events are Dependent

as $P \left(x | y\right) \ne P \left(x\right)$ we know that the events are Dependent