If you’re like many this weekend, you’ll do some gambling on the Super Bowl. Whether it’s a squares pool at a Super Bowl party, some prop bets in Vegas, or a mayoral contest between the chief executives of Baltimore and San Francisco (Rice-a-Roni against some DVDs of The Wire?), you’ll have opportunities to either win or lose based on probability. So here’s a tip that can help you on both football bets and the GMAT:
People are generally pretty bad at pairs probability.
Here’s an example – if you were to bet a friend on “will this year’s Super Bowl champion repeat as next year’s Super Bowl champion?”, your friend might see the *random* odds as 1/64 (since the GMAT only deals in random probability, we’ll take actual talent, coaching, contract status, draft position out of the equation!). That’s because, in order for the 49ers, say, to repeat, they’ll have to win this year’s championship (a 1/2 chance) and then next year’s championship (and they’re 1 team out of 32).
But this is wrong – your bet doesn’t ask for the probability of one *particular* team winning both Super Bowls, but rather the probability of “this year’s champion” (whichever team wins) doing it again next year. This year’s probability does not matter! Someone will win, and so you’re only concerned with that team’s (whatever it is – and there’s a 100% probability that there will be a winner) probability of repeating. Whatever that team is will have a 1/32 chance (again, just keeping it random) of repeating.
This is a concept that does get tested on the GMAT, and when it does there’s always a trap answer. Consider the question:
On three consecutive flips of a coin, what is the probability that all three produce the same result?
The trap answer here is 1/8 – you might look at this as a 1/2 probability on the first flip, then a 1/2 on the second, and a 1/2 on the third for a 1/8 probability, but remember – in this case the result of the first flip doesn’t have to be one or the other. Your job is just to match whatever the first result was on the next two. If the first was heads, then you need heads next (a 1/2 chance) and heads again (a 1/2 chance). And if it were tails, then you need tails (1/2) then tails (1/2). But because “any match will do” and you don’t care that it’s a specific match – the question doesn’t specify all heads or all tails, just “all of one of them” – your probability doubles because you’re not concerned about the result of the first event, you’re only concerned about matching whatever that result was.
So for probability questions that ask about pairs or matches, remember:
1) Check whether you need a *specific* pair/match or not.
2) If you don’t need a specific pair, but “any pair will do,” then the probability of the first result is 100% – something will happen.
3) If you need to guess, keep in mind that if it’s an unspecified pair/match, it’s almost certain that one of the trap answers will be a smaller number than the correct answer (in the above case, 1/8 is a trap and 1/4 is correct), so you can confidently rule out the smallest number and use number properties to try to eliminate another 1-2 answers.
Oh, and remember that your friends are probably pretty bad at pairs probability, too (nearly everyone is, especially after drinking a few of the products that will be advertised throughout the Super Bowl), so feel free to use the true pairs probabilities to your advantage on some prop bets.