Ok so I am obviously no math wiz and don't know how this works but let's say you have 100 of somthing we will go with marbles here. OK so you have 100 marbles and 45 are red while 55 are (Duke) blue. Obviously you got a 45% chance of drawing a red if you pull only one out without looking, but if you place it back what are the odds you draw red again? How about drawing a red 5 times in a row if you put it back everytime?
"The greatest enemy of knowledge is not ignorance, it is the illusion of knowledge" -Stephen Hawking
Once an event has happened, it can no longer impact a future event. So, each time you draw a red marble (if you then put it back), you are not impacting the odds that the next marble will be red. So, the first draw is a 45% chance, the second is a 45% chance, the third is a 45% chance and so on. So, if you have already drawn 4 reds, the odds that the 5th will be red is still 45%.
That said, if an event has not happened yet and you are trying to predict multiple outcomes, those odds are cumulative. So, if you have not made any draws and want to draw 5 reds in a row, your odds are (.45 x .45 x .45 x .45 x .45) = 1.84%.
-Jason "does that make sense?" Evans
I don't know what you are doing right now, but if you aren't listening to the DBR Podcast, you're doing it wrong.
The lawyer in me asks: "What does the client want the answer to be?"
Hmmm didn't realize it was that simple. So less than a 2% chance of the same things occuring 5 straight times?
If the original odds are 45%. Seems really low to me though.
"The greatest enemy of knowledge is not ignorance, it is the illusion of knowledge" -Stephen Hawking
It is low, but think about something as simple as flipping a coin.
One flip - probability of getting heads is 50%.
Two flips - probability of getting heads twice drops to 25%.
Three flips - probability of getting heads three times drops to 12.5%.
Four flips - probability of getting heads four times drops to 6.25%
Five flips - probability of getting heads five times drops to 3.125%
And your marble situation was less than 50% every time, so it's even lower.
Of course your marble situation would drop even lower if they did not replace the marble each time after it being drawn.
What about the hot hand?
Doyle Brunson would disagree too. A big part of his Super Syestems on hold-em is essentially the hot hand theory (or in his terms, a "rush" at the poker table).
brevity raises a good point -- the mathematical formula does not really apply to human actions. If someone is a 45% three point shooter, the formula does not really hold because focus, fatigue, confidence (or lack thereof), game pressure (or lack thereof), quality of the defense, etcetera do influence the outcome. It is not really random; past actions directly effect outcome. (Put another way, his shooting percentage is a description of past performance as opposed to a true odds to predict future outcomes).
Last edited by OldPhiKap; 05-20-2016 at 07:21 AM.
“That’s just, like, your opinion, man.”
You can always create a spreadsheet with trials based on the rand() function, and run it yourself just to get an empirical idea of what the closed form solution is.
(this was the final straw for me in terms of accepting the solution to the monty hall problem).
and yes, I know that random number generators are imperfect, etc., but it's close enough to demonstrate that the odds are indeed very low, so long as you're considering it at the start of the experiment. Thought of another way, what are the odds of NEVER picking blue in 5 draws, when there are 55/100 of them each time? They're hard to avoid.
Drawing marbles out of a bag or flipping a coin are independent events. Basketball shooting streaks are not. Probabilities are not calculated the same way when you cannot assume independence of events. Multiple hands of poker? Independent events. Who you play with? Not independent. There will be player effects to consider when calculating the odds of winning at poker. The odds of winning are different then the odds of the hands you are dealt.
The law of averages says that over a large enough number of events, the expected value will (basically) equal the actual value. So - in terms of the 45% shooter - you can't expect 45% success every game, but you can expect that over the course of a season. The law of averages does not say that just because you've gotten 5 heads in a row that tails are "due". I once tried to explain this concept to a sports betting friend. He asked, "If you flipped a coin 500 times and got 500 heads, what would you bet for the next flip?" My answer, which amazed him, was heads. Why? Because after 500 heads in a row, I no longer believe it's a fair coin.
A friend rails against this phrase. He always says, "it's just 'pressure'."
I think the phrase can sometimes be useful. A shooter who is hard on himself, expects such a great deal of himself, and gets nervous, may feel "pressure" even when taking a shot and his team is up 90 - 65 with a minute to go, but that feeling seemingly would not be described as "game pressure." Though I understand that concept and guess the phrase can be warranted, I tend to agree maybe just "pressure" would suffice.
"Score the basketball" ... "sports hernia" ... "game pressure" ... sometimes, we use two or three words when one would suffice: score, hernia, pressure.
Love the opportunity to take a statistics thread and turn it into a semantics pet peeves debate. I'm with you on the general point on redundant verbiage. Most of the time, adding the word "game" before "pressure" feels superfluous, because it's usually said while a game is happening. If you're talking in the abstract, and just say "He isn't really handling the pressure very well" then I guess it could be helpful to note the context. If we're talking about a college hoops player, there is pressure to perform in practice, there's pressure to perform in the classroom, there's pressure to perform in games. Those pressures are arguably different from one another. Of which type are we speaking? I can see the adjective of "game" being useful in that case.
"Score the basketball" drives me nuts. It isn't so much a lack of concision or superfluous words as it is just technically incorrect. You score a basket. A "basket" is, in addition to a noun synonymous with "hoop" (or if you're Ted Cruz, "ring" ) denoting a physical part of the game, a universally recognized unit of scoring. How many points does a team get for scoring a basketball? Nonsensical. In addition to bringing to mind an image of a guy crosshatching little slits in the ball with a pocketknife.
Other sports world redundancies: "a new record!" "pass the ball," "totally/absolutely/completely dominated/destroyed/whatever."
Never thought of the crosshatching image for "score" -- that fits. K and Jason Williams are two pretty bad offenders of "score the basketball" usage.
Add "a new tradition" to the sports-nonsense.
To bring this discussion back to its math origins, I believe the probability of nearly all these annoying things continuing (and language snoots continuing to feel offense) is roughly 100%.