Imagine that after picking on object, your friend looks at it (without you seeing the object) and tells you that the object is a square. Given this new information, you can completely ignore the fact that there were triangles. Instead, you can imagine that you were just picking from a collection of 1 white square and 2 black squares. This new information effectively lets you shrink your universe of possibilities to just those that are squares.
Given the information that you picked a square
Since you know that you picked a square, a white square is the only white object that you could pick. Therefore,
Here's another dumb question, but which will be relevant below: given that you know you picked a square, what is the probability you picked a square?
We write conditional probabilities using a bar $|$. We denote the probability of event $A$ conditioned on the fact that event $B$ occurred as $P(A\,|\,B)$. In this notation, the results are:
$P(B\,|\, S) = $
$P(W\,|\,S) = $
.
It turns out we can express a conditional probability (such as $P(B\,|\,S)$) in terms of the original probability of the combined event (in this case, $P(B,S)$) along with the probability of the additional information (in this case, $P(S)$). Let's explore the relationship between these three probabilities.
Before you knew that the object was a square, how many total objects were you picking from?
As each object was equally likely, the probability of picking any particular one of those $＿$ objects was
.
But, once you knew that you had picked a square, the universe of possible objects had shrunk to only
objects. With this new information, you could know that the probability you had actually picked any particular one of those $＿$ remaining objects had increased to
.
This increase in probability is directly related to the probability of picking a square. Before your friend gave you any information about the object, what was the probability that the object was a square? $P(S) = $
. Once your friend informed you that you had a square, the probability of a square increased to
. This increase in probability corresponds to
the probability $P(S)=＿$.
This division by $P(S)$ captures exactly how probabilities increase upon the knowledge that the object is a square, as the information decreases the universe of possible outcomes down to a set of outcomes that originally occurred with probability $P(S)$. Indeed, if you take the original probability per object of $＿$ and divide by $P(S)=＿$, you get
, which should match your previous calculation for the probability for any particular one of the $＿$ objects that were squares.
Verify that the probabilities of getting a black or white object conditioned on obtaining a square follow this pattern.
$P(B\,|\,S) = P(B, S)/P(S) = $
$/$
$=$
$P(W\,|\,S) = P(W, S)/P(S) = $
$/$
$=$
One more way to see the effect of conditioning on probabilities is through the contingency table. One you receive the information that the object was a square, the universe of possible outcomes shrinks to the row corresponding to the square. Since we know the row had to occur, its total probability must increase to
. Dividing by the original total probability for that row, $P(S)=$
, the contingency table, conditioned on the fact that the object was a square, becomes the following table. (Fill in this table with numbers for the conditional probabilities.)
Imagine, instead, that your friend told you that the object was white. What are the probabilities of the object being a square or a triangle conditioned on that information? You can take the column of the contingency table corresponding to white objects and rescale it so that the total is $1$ (given that the allowable universe of outcomes are just those in that column).
Fill in the numbers for this rescaled contingency table.
What quantity did you divide the column by to rescale it? (Enter the mathematical expression for the probability, not the actual number itself.)
The first two numbers of the contingency table were $P(S\,|\,W)$ and $P(T\,|\,W)$. Write the formula for these conditional probabilities in terms of the expressions for the original probabilities for the events (such as $P(B,S)$ or $P(W,T)$) along with the expression $＿$ you used to rescale these quantities.
$P(S\,|\,W) = $
$= ＿$
$P(T\,|\,W) = $
$= ＿$
Calculate the probability of the object being black or white conditioned on the information that the object was a triangle. For each conditional probability, enter a mathematical expression (in terms of quantities like $P(W,T)$, etc.) in the first blank and enter a number in the second blank.
$P(B\,|\,T) =$
$=$
$P(W\,|\,T) =$
$=$