Are the lottery winning numbers truly random?
If the lottery is not crooked, yes.
There are two types of random numbers: TRN's and PRN's. True Random Numbers and Pseudo Random Numbers.
Pseudo Random Numbers are generated by computer and when the same seed value is used the algorithm will generate the same sequence of numbers which will at some point repeat. The repetition can be short or very long depending on the algorithm used. Most PRN's use derivatives of an algorithm called LFSRs (Linear Feedback Shift Registers). The most well known/widely used algorithm today is called the Mercene Twister. Some early online casino's were compromised because a sequence could be predicted if one knew the algorithm used and had a short sequence of the numbers that were sequentially generated. So today online casino's make a lot more effort to generate unduplicatable PRN combinations.
Lottery numbers are True Random Numbers. This means to predict the numbers one would have to either simulate the universe up to the point where the next lottery numbers are drawn, or one would have to know the EXACT initial conditions just before the numbers are drawn and the exact mechanical mechanism used so that the process can be simulated using cellular automata or a pure physics simulation.
Examples of TRNs are lottery numbers, radio static and cosmic radiation values.
Flipping a coin in the long run you'll have 50/50 heads and tails.
What does long run mean? 10 times? 100? 1000? 1000 000 000 000?
This is referred to as the gamblers fallacy.
The inverse of this is called the hot hand or inverse gamblers fallacy.
See the paper available online: Predicting Lotto Numbers by Claus Bjørn Jørgensen, Sigrid Suetens, and Jean-Robert Tyran
From the paper: "Mounting evidence from the experimental laboratory and the field suggests that truly random processes are difficult to grasp for most people, and that many people tend to see patterns in data when in fact there are none. According to the “law of small numbers” (Tversky and Kahneman, 1971), people tend to mispredict random sequences because they expect small samples to “look like” large samples. For example, if asked to generate a random sequence as in a coin toss, many people predict too many switches between head and tails because they falsely believe that head and tails should appear in equal proportion even in a small sample (see, e.g., Bar-Hillel and Wagenaar, 1991; Rapoport and Budescu, 1997). Or when playing the roulette, people tend to expect that a black number is “due” after observing a sequence of red numbers (e.g. Croson and Sundali, 2005). The belief in frequent reversals in random sequences has been dubbed the “gambler’s fallacy” (Tversky and Kahneman, 1971)."
I could spew a lot more about this but yeah...