Kailassa
Originally posted by Kailassa
You're being dealt playing cards, and there are only 4 types of card, ace, king, queen and jack.
You want 16 cards in a particular order.
You have 1 chance in 4294967296 of getting all the cards in the right order. This is the cumulative probability.
Why do so many people apparently have so much difficulty in working out simple combinations and probabilities ?
In your above example, you haven't specified any particular order such as A,A,A,A,K,K,K,K,Q,Q,Q,Q,J,J,J,J or J, Q, K, A, J, Q, K, A, J, Q, K, A, J,
Q, K, A, etc and so I'm going to assume that ANY order will do.
If that's the case then with 16 cards only, there are exactly 20,922,789,888,000 different ways of arranging those 16 cards ... and
NOT
4,294,967,296 as you've stated.
So this means that if you
do want just one particular order of those 16 cards, then you have just 1 chance in 20,922,789,888,000 of dealing the
cards and
actually getting the order that you want.
When you have the first 15 cards in place the odds of getting the queen you want are not 1 in 10^9, they are 1 in 4. The odds for getting any single
correct card are 1 in 4. Your notion that the cumulative probabiliy should be applied to each card draw, and then all multiplied together, is just
nonsense.
Again, you haven't quite got a handle on cumulative probability (for multiple sequential draws) as opposed to individual probability (for a single
draw).
Let me try to make it clearer for you by using a normal (unbiased) coin in the following examples.
Ok, for the 1st experiment, all we're interested in is tossing the coin and getting a head.
What's the odds (probability) of actually getting that head ? Of course it's just 1 in 2 or P=0.5.
Now we're going to toss that same coin and this time hope to get a tail.
What's the odds of a tail this time ? Again, it's just 1 in 2 or P=0.5.
Now we're going to toss that coin again a 3rd time and hope to get a head.
What's the odds of a head appearing on this 3rd toss ? No rocket science here as again, it's just 1 in 2 or P=0.5.
The important thing to bear in mind here is that
we don't care at all what the previous tosses gave as each one of the 3 tosses is treated as
a
UNIQUE and isolated toss ... each toss completely independent of past or future tosses.
This means that no matter how many tosses you make, 1, 2, 20, 100, 1000, etc, each toss has exactly the same probability as any other toss i.e. P=0.5
when treated as
completely independent of any past/future tosses.
Now, in this 2nd coin tossing experiment, and before we toss any coins, we decide in
advance that we want to see the following result after 2
tosses ... a HEAD followed by a TAIL.
Unlike the 1st experiment, this time we're not interested in individual probabilities but rather the combined or
CUMULATIVE probability of
successfully achieving our goal, i.e a H followed by a T.
So, this time we have to multiply together the individual probabilities to get a cumulative probability. In other words, the 1st toss has a P=0.5 ...
the 2nd toss has a P=0.5 ... but the cumulative probability is calculated as
0.5 x 0.5 = 0.25.
This means that you have only P=0.25, or 1 chance in 4 of getting a H followed by a T.
To show this is correct, here are the 4 possible outcomes after tossing a coin twice:
HH
HT
TT
TH
As can be seen, the desired outcome of a H followed by a T is just 1 possibility out of a total of 4 ... or in other words, it has a P=0.25 ... which
is what was the result of our
cumulative probability calculation just above.
So the way that you looked at it was from an independent pick/toss point of view, which is fine if we're only interested in a single outcome e.g. a H
or a T or a K or a Q, etc ... but we're actually interested in multiple picks resulting in just one
specific predetermined outcome ... which
in this case happens to be a final 153 picks (random mutation) of the correct nucleotides.
So determining the cumulative probability of nature stringing together the correct sequence of 153 nucleotides (the "expected" outcome) is exactly
the same scenario as deciding what coin sequence you want to see but only
before you've started the tossing.
And the cumulative probability is 0.25^153 ... with the odds stacked astronomically
AGAINST the correct 153 nucleotides being strung together
in the required sequence somewhere in a chromosome.
Hope the above has clarified your misunderstandings regarding simple combination and probability theory
TheWill
Originally posted by TheWill
reply to post by tauristercus
No, identical mutations aren't expected to occur simultaneously in two separate organisms.
but
... the number of organisms within a system is VERY relevent when talking about the odds. One of them that receives a beneficial mutation in the germ
line(and just so you know, most proteins are likely to have evolved in very rapidly-reproducing organisms, which produce hundreds or even thousands of
babies at any one time, rather than slow reproducing tetrapods which rarely go over a couple of hundred in their lifetimes) would enjoy greater
survival of their offspring which inherited the mutation, and so on and so on, until they outnumbered their conspecifics without the mutution, and
were more likely to mate with a relative than a non-relative, until eventually the allele comes to fixation in the population.
Ok, I can see now that we're looking at this from 2 different points of view.
Your point of view, based on your above examples, is that you're essentially assuming that any mutation, as long as it doesn't prevent the owner
from breeding successfully, will eventually become assimilated into the species genome. And this I tend to assume would apply to not only completely
functioning proteins but also for nucleotide sequences that don't currently result in a functioning protein but with additional fortuitous mutations,
potentially
might result in a working protein somewhere in the future.
I however am looking at it from the point of view that even if there were millions of copies of partial (potential proteins of the future) nucleotide
sequences, that still does
not alter the fact that at best one and only one organism in its species will receive the final chunk of nucleotide
sequence through random mutation to be capable of expressing that protein.
Only then will there be a possibility of passing on the complete working nucleotide sequence to its descendants and locking it into the species
genome.
I'll try to make it clearer ... just in case I've confused everyone even more
Again, I'll be using my favourite protein, namely insulin.
Firstly, can we agree on a few of initial points ?
1. At some point in evolutionary history, insulin did
not exist in any organism/specie.
2. There is very little chance whatsoever that a single mutation gave rise to the necessary (and correct) 153 nucleotide sequence in any organism.
3. That the insulin sequence was initially started, and then gradually added to, over evolutionary time.
Ok, let's assume that at some point in time an organism, gained through random mutations, say the 1st 20 correct nucleotide sequences.
This organism was a fast breeder and eventually this 20 sequence strand became established in the genome ... therefore many, many copies of this
sequence will exist.
Now at this point, you'd think with so many copies, that nature would now have many more opportunities for additional random mutations of this
sequence and that it would be so very much easier to complete the entire sequence and arrive at the insulin gene.
But here's the point that's been missed.
It makes no difference if there is just one copy of that 20 base sequence or millions of copies of that 20 base sequence. From a probability point of
view, all these millions of copies are treated as essentially just
ONE copy because the odds of a random mutation inserting the next correct
nucleotide into position 21 of the sequence is
IDENTICAL for each of those millions of copies.
Eventually, one of those copies may receive that correct 21st mutation at which point, every other 20 sequence is now redundant and we have now gone
from millions of identical copies to just a single "enhanced" copy in just one individual organism.
Then the cycle starts again ... that single organism needs to breed profusely to establish the new 21 sequence strand into the genome ... one of those
millions of descendant organisms will have to wait for additional mutations adding to that 21 sequence strand resulting in once more, just a single
organism carrying the new sequence ... breeds ... etc ... etc ... etc
Therefore, if the odds of one of those organisms with the 1st 20 nucleotide sequence in place is say (just an example) 1 in 10^50 of acquiring the
remaining 133 nucleotides in the correct sequence to make insulin ... then the
exact same odds apply to
every other organism that
carries those same initial 20 nucleotides.
So 1 organism or 1 million organisms carrying the identical 20 sequence, the odds remain the same for each of them ... absolutely astronomical !
So from natures point of view, all the multiple copies are completely superfluous and the evolution of the insulin sequence from 1 nucleotide to the
final 153 nucleotides, for all intents and purposes, only takes place in just one organism at a time. Just like pruning back every branch on a tree
except one and then following the path of that one single branch as it develops over time whilst continuously pruning away any side side branches as
they develop on that branch we're following.