All of probability in one line : Take an (interesting) probability space $(\Omega, p)$ and an (interesting) event $A\subseteq \Omega$. Find $\mathbf{P}(A)$.
This is the mathematical side of the picture. It is easy to make up any number of probability spaces - simply take a finite set and assign non-negative numbers to each element of the set so that the total is $1$.
There are $2^{\#\Omega}=2^{2^{n} }$ events. One example is $A_{k}=\{\underline{\omega}{\; : \;} \underline{\omega}\in \Omega \mbox{ and } \omega_{1}+\ldots +\omega_{n}=k \}$ where $k$ is some fixed integer. In words, $A_{k}$ consists of those $n$-tuples of zeros and ones that have a total of $k$ many ones. Since there are $\binom{n}{k}$ ways to choose where to place these ones, we see that $\#A_{k}=\binom{n}{k}$. Consequently, $$ \mathbf{P}\{A_{k}\}=\sum_{\underline{\omega}\in A_{k} }p_{\underline{\omega}} = \frac{\# A_{k} }{2^{n} }=\begin{cases} \binom{n}{k}2^{-n} &\mbox{ if }0\le k\le n, \ 0 & \mbox{ otherwise}. \end{cases} $$ It will be convenient to adopt the notation that $\binom{a}{b}=0$ if $a,b$ are positive integers and if $b > a$ or if $b < 0$. Then we can simply write $\mathbf{P}\{A_{k}\}=\binom{n}{k}2^{-n}$ without having to split the values of $k$ into cases.
The cardinality of $\Omega$ is $m^{r}$ (since each co-ordinate $\omega_{i}$ can take one of $m$ values). Hence, if we set $p_{\underline{\omega}}=m^{-r}$ for each $\underline{\omega}\in \Omega$, we get a valid probability space.
Of course, there are $2^{m^{r} }$ many events, which is quite large even for small numbers like $m=3$ and $r=4$. Some interesting events are $A=\{\underline{\omega} {\; : \;} \omega_{r}=1\}$, $B=\{\underline{\omega}{\; : \;} \omega_{i}\not=1 \mbox{ for all }i\}$, $C=\{\underline{\omega}{\; : \;} \omega_{i}\not=\omega_{j} \mbox{ if } i\not= j\}$. The reason why these are interesting will be explained later. Because of equal elementary probabilities, the probability of an event $S$ is just $\#S/m^{r}$.
Thus, $\mathbf{P}(C)=\frac{m(m-1)\ldots (m-r+1)}{m^{r} }$. Note that this formula is also valid for $m < r$ since one of the factors on the right side is zero.
In real life, there are often situations where there are several possible outcomes but which one will occur is unpredictable in some way. For example, when we toss a coin, we may get heads or tails. In such cases we use words such as probability or chance, event or happening, randomness etc. What is the relationship between the intuitive and mathematical meanings of words such as probability or chance?
In a given physical situation, we choose one out of all possible probability spaces that we think captures best the chance happenings in the situation. The chosen probability space is then called a model or a probability model for the given situation. Once the model has been chosen, calculation of probabilities of events therein is a mathematical problem. Whether the model really captures the given situation, or whether the model is inadequate and over-simplified is a non-mathematical question. Nevertheless that is an important question, and can be answered by observing the real life situation and comparing the outcomes with predictions made using the model 2 Roughly speaking we may divide the course into two parts according to these two issues. In the probability part of the course, we shall take many such models for granted and learn how to calculate or approximately calculate probabilities. In the statistics part of the course we shall see some methods by which we can arrive at such models, or test the validity of a proposed model. .
Now we describe several ''random experiments'' (a non-mathematical term to indicate a ''real-life'' phenomenon that is supposed to involve chance happenings) in which the previously given examples of probability spaces arise. Describing the probability space is the first step in any probability problem.
The corresponding probability model : Since there are two outcomes, the sample space $\Omega=\{0,1\}$ (where we use $1$ for heads and $2$ for tails) is a clear choice. What about elementary probabilities? Under the equal chance hypothesis, we may take $p_{0}=p_{1}=\frac{1}{2}$. Then we have a probability model for the coin toss.
If the coin was not fair, we would change the model by keeping $\Omega=\{0,1\}$ as before but letting $p_{1}=p$ and $p_{0}=1-p$ where the parameter $p\in [0,1]$ is fixed.
Which model is correct? If the coin looks very symmetrical, then the two sides are equally likely to turn up, so the first model where $p_{1}=p_{0}=\frac{1}{2}$ is reasonable. However, if the coin looks irregular, then theoretical considerations are usually inadequate to arrive at the value of $p$. Experimenting with the coin (by tossing it a large number of times) is the only way.
There is always an approximation in going from the real-world to a mathematical model. For example, the model above ignores the possibility that the coin can land on its side. If the coin is very thick, then it might be closer to a cylinder which can land in three ways and then we would have to modify the model...
Thus we see that example 9 is a good model for a physical coin toss. What physical situations are captured by the probability spaces in example 10 and example 11?
Example 10 : This probability space can be a model for tossing $n$ fair coins. It is clear in what sense, so we omit details for you to fill in.
The same probability space can also be a model for the tossing of the same coin $n$ times in succession. In this, we are implicitly assuming that the coin forgets the outcomes on the previous tosses. While that may seem obvious, it would be violated if our ''coin'' was a hollow lens filled with a semi-solid material like glue (then, depending on which way the coin fell on the first toss, the glue would settle more on the lower side and consequently the coin would be more likely to fall the same way again). This is a coin with memory!
Example 11 : There are several situations that can be captured by this probability space. We list some.
In that example, $A$ is the event that the last ball ends up in the first bin, $B$ is the event that the first bin is empty and $C$ is the event that no bin contains more than one ball.
The next example is more involved and interesting.
A possible probability model : Let there be $n$ genes in each human, and each of the genes can take two possible values (Mendel's ''factors''), which we denote as $0$ or $1$. Then, let $\Omega=\{0,1\}^{n}=\{\mathbf{x}=(x_{1},\ldots ,x_{n}){\; : \;} x_{i}=0 \mbox{ or }1\}$. In this sense, each human being can be encoded as a vector in $\{0,1\}^{n}$.
To assign probabilities, one must know the parents. Let the two parents have gene sequences ${\bf a}=(a_{1},\ldots,a_{n})$ and ${\bf b}=(b_{1},\ldots ,b_{n})$. Then the possible offsprings gene sequences are in the set $\Omega_{0}:=\{\mathbf{x}\in \{0,1\}^{n}{\; : \;} x_{i}=a_{i} \mbox{ or }b_{i}, \mbox{ for each }i\le n\}$. Let $L:=\#\{i{\; : \;} a_{i}\not=b_{i}\}$.
One possible assignment of probabilities is that each of these offsprings is equally likely. In that case we can capture the situation in the following probability models.
Reasonable model? There are many simplifications involved here. Firstly, genes are somewhat ill-defined concepts, better defined are nucleotides in the DNA (and even then there are two copies of each gene). Secondly, there are many ''errors'' in real DNA, even the total number of genes can change, there can be big chunks missing, a whole extra chromosome etc. Thirdly, the assumption that all possible gene-sequences in $\Omega_{0}$ are equally likely is incorrect - if two genes are physically close to each other in a chromosome, then they are likely to both come from the father or both from the mother. Lastly, if our interest originally was to guess the eventual height of the child or its intelligence, then it is not clear that these are determined by the genes alone (environmental factors such as availability of food etc. also matter). Finally, in case of the problem that Solomon faced, the information about genes of the parents was not available, the model as written would be use.