[quote name=\'mxc0427\' post=\'193998\' date=\'Aug 14 2008, 01:18 AM\']
Scenario 1 is that you have NO IDEA to the answer. With the 50:50, you are left with a 50% chance (1/2) of getting the question right. Double Dip allows two guesses. The first guess is out of a possible 4 answers. Thus 25% (1/4). Getting the question wrong will yield a 1/3 guess (33%). In statistics, AND = multiplication and OR= addition. Because you have the chance of getting the question correct on the first attempt OR the second attempt, adding the probabilities will give you a 58% of answering the question correctly.
Looking at the other way with the Double Dip, you have a 75% (3/4) of getting the question wrong on the first attempt and a 66% (2/3) of getting the question wrong on the second attempt. Because answering the question wrong requires an incorrect answer on the first attempt AND the second attempt, you multiply these probabilities to give you about a 50% of missing the question. So as you can see, you have the same failure percentage for both lifelines, but a higher probability of getting a question right with Double Dip.
Scenario 2 is when you can knock out one of the answers as being incorrect in your head... When the Double Dip is issued in this scenario, the first attempt you have at the question is 33% (1/3). Getting it wrong on the first try would yield a 50% on the next attempt (1/2). Again, you either get the question right on your first attempt OR the second attempt. So with this method, you have increased your odds to 83%. With having a gut instinct on one of the answers being incorrect, the odds for the 50:50 remain the same, but the Double Dip increases by 25% (because you eliminated one of the four possible choices.)
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The spirit of your calculations is correct, but there's a problem. In scenario 1, you say there's a 58% chance of being right with Double Dip, and a 50% chance of being wrong. That can't happen (the total is 108%). The error is in scenario 1, where you add 25% and 33%. The probability of being correct the first time *is* 25%, and the probability of being correct the second time *is* 33% -- but you won't be correct the second time if you were already correct the first time. The calculation should be based on being correct the first time, OR (being incorrect the first time AND being correct the second time), which gives:
(1/4) + (3/4)(1/3) = 1/2 = 50%
Even simpler is to realize that with Double Dip, you get to choose 2 answers out of 4... and as deceptive as probability can be sometimes, that just works out to be 2/4 = 1/2.
There's a similar issue with the 33% + 50% calculation in scenario 2. It should be:
(1/3) + (2/3)(1/2) = 2/3 = 67%,
or, again, you can simply say you get two choices out of the three answers.
Also, 50-50 leaves the same 67% probability in scenario 2. Let's assume without loss of generality that A is right, but that you can rule D out. Under 50-50, you would be left with either AB, AC, or AD, so 2/3 of the time you would still have to guess. Thus, the probability of being right is:
P(having AD) + P(not having AD)*P(guessing correctly) = (1/3) + (2/3)(1/2) = 2/3 = 67%.
However, here is a third scenario:
Scenario 3: You can narrow it down to two answers in your head. With Double Dip, you will certainly get the correct answer; if you were wrong the first time, you'll be right the second time. However, with 50-50, there is still a chance you'll be wrong -- if A was right and you ruled out C and D, you could still miss the question if 50-50 left you with AB.
So the two lifelines are the same if you're clueless or can only rule out 1 answer; Double Dip is better if you can rule out 2 answers.
/ruling out 3 answers is left as an exercise to the reader
