Subsection 1.7.1 Probability terminology (Ch 13–15)
[Note: The symbols given in this section appear in Friedman’s text in a note on p.227 at the end of Exercise Set B.]
In a box model, an outcome is a sequence of tickets obtained by random draws, either with or without replacement. For example, here are the 12 possible outcomes for 2 draws, without replacement, from the box \(\{A,K,Q,J\}\text{.}\) (This is a box model for "two cards are dealt from a 4-card deck containing an ace, a king, a queen, and a jack".)
\begin{equation*}
\text{set of all outcomes} = \{AK,AQ,AJ,KA,KQ,KJ,QA,QK,QJ,JA,JK,JQ\}
\end{equation*}
In this notation, \(AK\) means the ace is drawn first, and the king second, while \(KA\) means the king is first and the ace is second.
An event is a set of outcomes. For example, the events "get a queen" and "get a king on the first draw" are
\begin{align*}
\text{"get a queen"} \amp = \{QA,QK,QJ,AQ,KQ,JQ\}\\
\text{"get a king on draw 1"} \amp = \{KA,KQ,KJ\}.
\end{align*}
Another example is the event "get the hand ace-king". In a "hand", it does not matter what order the cards are dealt in, so we have
\begin{equation*}
\text{"get the hand ace-king"} = \{AK,KA\}.
\end{equation*}
To say that two events \(E\) and \(F\) "both happen" can be rephrased \(\text{"}E \AND F\text{"}\text{.}\) To say that "at least one of the events \(E,F\) happens" can be rephrased \(\text{"}E \OR F\text{"}\text{.}\) For example, for the events
\begin{gather*}
E = \text{"get an ace"} = \{AK,AQ,AJ,KA,QA,JA\}\\
F = \text{"get a king"} = \{KA,KQ,KJ,AK,QK,JK\}
\end{gather*}
we have the following.
\begin{align*}
E \AND F \amp = \text{"get a hand with an ace and a king"} = \{AK,KA\}\\
E \OR F \amp = \text{"get a hand with an ace or a king (or
both)"}\\
\amp = \{AK,AQ,AJ,KA,QA,JA,KQ,KJ,QK,JK\}
\end{align*}
[Comment for students who know about intersection and union of sets: the event \(\text{"}E \AND F\text{"}\) is the same thing as the intersection of the sets \(E,F\text{,}\) and the event \(\text{"}E \OR F\text{"}\) is the same thing as the union of the sets \(E,F\text{.}\)]
We will use the correct terms "outcome" and "event" instead of the vague term "thing" used in the text in Ch 13.
We will write \(P(E)\) to denote the probability for an event \(E\text{.}\) For example, in the game of 2 draws, with replacement, from the box \(\{A,K,Q,J\}\text{,}\) the probability of the single outcome \(AK\) is
\begin{equation*}
P(AK) = 1/12 \approx 8.3\%
\end{equation*}
because there are 12 equally likely outcomes for the 2 draws. The probability for getting the hand ace-king is
\begin{equation*}
P(\text{get the hand ace-king}) = 2/12 \approx 16.7\%
\end{equation*}
because there are 2 equally likely outcomes in the event "get the hand ace-king", and there are 12 equally likely outcomes in all for the 2 draws. The probability of getting a queen is
\begin{equation*}
P(\text{get a queen}) = 6/12 = 50\%
\end{equation*}
because there are 6 equally likely outcomes in the event "get a queen". The probability of getting king on the first draw is
\begin{equation*}
P(\text{get a king on draw 1}) = 3/12 = 25\%
\end{equation*}
because there are 3 equally likely outcomes in the event "get a king on draw 1".
The textbook does not give a definition of the term conditional probability; instead, they explain by example in Ch 13 Sec. 2 (p.226). Here is the definition: The conditional probability of event \(E\) given that event \(F\) has happened, denoted \(P(E|F)\text{,}\) is
\begin{equation}
P(E|F) = \frac{P(E \AND F)}{P(F)}\tag{1.7.1}
\end{equation}
where the event "\(E \AND F\)" is the set of all outcomes that belong to both events \(E\) and \(F\text{.}\) For example, in 2 draws with replacement from the box \(\{A,K,Q,J\}\text{,}\) we have
\begin{equation*}
P(\text{get a queen | get a king on draw 1}) = 1/3 \approx 33.3\%
\end{equation*}
because there is 1 outcome, namely \(KQ\text{,}\) that has a \(Q\text{,}\) among the 3 outcomes that have \(K\) on draw 1. We also have
\begin{equation*}
P(\text{get a king on draw 1 | get a queen}) = 1/6 \approx 16.7\%
\end{equation*}
because there is 1 outcome, namely \(KQ\text{,}\) for which draw 1 is a \(K\text{,}\) among the 6 outcomes that have a \(Q\text{.}\)
The definitions of independent and dependent events (box on p.230) should be replaced by the following: "Two events \(E,F\) are independent if \(P(F|E) = P(F)\text{.}\) Otherwise, the events \(E,F\) are dependent." Or in words, "Two events \(E,F\) are independent if the probability of \(F\) given that \(E\) has happened is equal to the probability of \(F\text{.}\) Otherwise, the events \(E,F\) are dependent."
