I am reading lecture #6 on the first volume of "feynman's lectures on physics", and I understood quite well the first half of the lecture. However after he proved that D(rms)= √D^2= √N, I started to lose him, and so I have quite a few questions: 1) "The variation of Nh from its expected value N/2 is Nh-N/2=D/2. The rms deviation is (Nh-N/2)rms=√N/2" I understand that the expected value for the difference between the number of heads and tails (AKA the distance traveled) is √N, and that the expected value for the number of heads is N/2. But how from these 2 facts did he conclude that the number of heads is expected to deviate from N/2 by √N/2 is what I do not understand. And further more, is that not a paradoxical conclution that on one hand you expect the number of heads to be N/2, and on the other hand you expect the difference between the number of heads and tails to be √N? 2) "According to our result for Drms, we expect that the “typical” distance in 30 steps ought to be √30=5.5, or a typical k should be about 5.5/2=2.8 units from 15. We see that the “width” of the curve in Fig. 6–2, measured from the center, is just about 3 units, in agreement with this result." By "typical" does he mean the average? And I don't understand the "width of the curve measured from the center" part. what does he mean by the center? center of what? and what does the width of the curve from said center tell us? how does it confirm our findings? 3) "We also expect an actual Nh to deviate from N/2 by about √N/2, or the fraction to deviate by 1/N*√N/2=1/2√N" Again, how did he come to that conclusion? It seems that he devided the deviation by N because Nh is also devided by N to reach the fraction. But does deviding the expected result of an experiment by a constant also means that the deviation will also be devided by said constant? 4) "Let us define P(x,Δx) as the probability that D will lie in the interval Δx located at x (say from x to x+Δx). We expect that for small Δx the chance of D landing in the interval is proportional to Δx, the width of the interval. So we can write P(x,Δx)=p(x)Δx." The 2nd sentence confuses me. I understand that the chance of D landing in an interval will grow proportionally to the size of the interval, the bigger the interval, the bigger the chance. But I do not understand what is the meaning of the last equation: The probability of D being in interval Δx= the probability of X times the interval? I know p(x) is the probability density function but I don't understand he derived it from this equation. 5) "We plot p(x) for three values of N in Fig. 6–7. You will notice that the “half-widths” (typical spread from x=0) of these curves is √N, as we have shown it should be." This is probably a result of me not understanding what I asked in question 2, but what does he mean by the "typical spread from x=0", for example in N=10,000 the half width of the curve is 100 near the top, so how is it typical? 6) "You may notice also that the value of p(x) near zero is inversely proportional to √N" how can the value of p(x) near 0 be inversly proportional to √N if N is a constant? and if he means that the curve gets wider as N gets bigger, why is it inversly proportional to √N and not just N? "Since the curves in Fig. 6–7 get wider in proportion to √N, their heights must be proportional to 1/√N", again, why not just N? Well I asked more than I planned, so if you actually take the time to answer all these you have my deepest gratitude.