+GreshamCollege Thank you for the lecture. You said you couldn't think of an easy way to prove the birthday paradox with the lecture hall. May I suggest you hand round a year's calendar and ask everyone to put a dot on their birthday while you carry on with the lecture. Assuming everyone takes 10s to do so it would take just over half an hour. If you wanted it to start when you started talking about the birthday problem and finish when you were ready to move on (around 6 mins), you could print the calendar on 5 transparencies and pass them along the rows with attached marker pens asking people to put a dot in a random position in the square indicating their birthday. Would be interesting to then calculate how likely it was to have the distribution of birthdays found on that particular occasion in the hall.
Alternative (in the gambling game , 23 min) finding probabilities of winning the game and the probablity of the complement event (probability of not winning the game ). Instead of counting YYYY and YYYM cases as equally likely outcomes, and because in reality for some outcomes there is no need to take the third or the forth trials (already it would be known that one of the players already won). Therefore we can calculate the probability of just winning by the following events: MM ( 0.5^2), MYM (0.5^3), YMM (0.5^3), , MYYM, YYMM, YMYM (each outcome with 0.5^4) , then by adding them together ( P(Win) = 0.6875), while probability of losing will be 0.3125.
What is interesting is that; if probability outcome is "limitted" to a given results say 1 regardless of evolutionary processes or mechanism, then the future of such results can no longer be judged randomly , but judged with predictable degree of certainty. And probability disappear with limitations.
The part with the IMHO most important implications starts at @25.11- the random walk. Especially the example with the school children. I wish more people would understand such aspects rather than all those boring games examples. Yes, that's how the math for it began, but it's also, I think, why only lovers of math care and the rest tune out - "why would I care?".
Looks more like { 365! / (365-n)! } / { 365 ^ n } use induction to prove it ;o) Of course : 365! is maddeningly huge , and the expression i use here is simply shorthand for (365-n+1) * (365-n+2) * ... * 365
Does the Birthday Problem presume a fair distribution of birthdays across the year? (Because, in the US at least, there are significantly more-than-expeceted births in July thru October, and less than expected during the other months.)
Yes, it does indeed assume an equal distribution. See for example Blitzstein's lecture on this: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-LZ5Wergp_PA.htmlm56s
Randomness is dual to order Certainty is dual to uncertainty -- The Heisenberg certainty/uncertainty principle Subjective is dual to objective Thesis is dual to anti-thesis, the Hegelian dialectic Energy is dual to mass -- Einstein Dark energy is dual to dark matter Energy is duality, duality is energy!
philosophical brain-workout : try to figure out what limsup _ { m -> +inf} liminf _ { n -> +inf } limsup _ { r -> +inf } f(p,q,r) means intutively where f(p,q,r) is a function from {the set of nonnegative integers}^3 to nonnegative integers.
