# Events being independent of one another

I'm having trouble interpreting a question.

Question: Given a table with four columns, like this: sorry I've to paste an image since I can't draw tables here on this site

Based on the data shown in the table above, is there evidence to suggest that whether people have a college degree is independent of whether they listen to radio news?

My interpretation:

According to me, this means that the probability of having a college degree is the same no matter whether you listen to radio or not (hence "independent of listening to radio").

Hence, the probability of having a college degree and listening to radio should be the same as that of having a college degree and not listening to radio.

I am interpreting it in the sense that if "X is independent of Y" then P(X and Y) is the same as P(X and not Y) - since it does not affect X, whether or not Y will occur.

However, KA has stated in their explanation that:

In this case, to be independent means that the probability that someone with a college degree listens to radio is approximately the same as the probability that someone without a college degree listens to radio.

That interprets as P(X and Y) = P(not X and Y), contrasting my previous conclusion.

Where is my train of thought going wrong?

Or is this English structure of "event X is independent of event Y" similar to that of "X times as many of Y as of Z" (that implies Y=XZ) i.e. it is just a rule and I've to mug it up?

• The definition of Independence is that $P(A\cap B)=P(A)\times P(B)$. Here, with $A$ being "college" and $B$ being "radio" we see that in our sample $P(A\cap B)=\frac {115}{442}\approx .26$ and $P(A)\times P(B)=\frac {432}{1600}\times \frac {170}{1600}\approx .0287$ so, even allowing for sampling error, they are nowhere close. – lulu Oct 5 '17 at 11:36
• @lulu Ah, that's reasonable as well. But, then what about the clashing interpretations of logic - mine vs KA? – Gaurang Tandon Oct 5 '17 at 12:03
• Your interpretation is not correct. Consider a throw of two independent dice. Let $A$ be the event "the first die comes up $1$" and let $B$ be the event "the second die comes up $5$" . These are clearly independent. But according to your approach we should compare The probability of both occurring ($\frac 1{36}$) to the probability that $A$ and $B^c$ both occur, ($\frac 5{36}$). Unless I am misreading what you wrote. – lulu Oct 5 '17 at 12:13
• Firstly, the seemingly contradictory statements just reflect the fact that independence is symmetric: If $A$ and $B$ are independent, then so are $B$ and $A$, similarly with their complements and so on. In that sense, both interpretations are fine. But, secondly, your formalisations are not correct: You want to think of this in terms of conditional probabilities rather than intersecting events: The question is $P(A|B)=P(A|B^C)$ (or $P(B|A)=P(B|A^C)$) which, I feel, is easier to interpret than $P(A\cap B)=P(A)P(B)$. Your nice first sentence ("According to me...") reflects just that. – Mau314 Oct 5 '17 at 12:45
• And to help with the solution: This exercise asks you to apply the "Fisher Test" en.wikipedia.org/wiki/Fisher%27s_exact_test - try that! – Mau314 Oct 5 '17 at 12:47