Kyle Hausmann is a Content Developer at Knewton.
If the background sound of your dreams over the last month has been the endless blowing of vuvuzelas, then you either have a really annoying neighbor or you are caught up in World Cup fever. At Knewton, we thought we could tie some of the football fanfare to probability. (Because that’s what we do here.)
So, here is a data sufficiency example:
If the probability is 0.7 that Uruguay will win any given match, and if the two semi-final matches are Uruguay vs. the Netherlands and Germany vs. Spain, what is the probability that Germany will win the final?
- The probability that Germany beats another European team in the semi-finals or in the final is 0.6.
- If they make it to the final, the probability that the Netherlands wins the Cup is 0.4.
The question assumes that the outcome of each game is independent from the others. Since we have independent events, we can multiply the probabilities of each event together.
First, we consider Statement 1 alone. With Statement 1, we know the probability (0.6) that Germany will beat Spain, a European team, and make it into the final. And from the prompt, we know the probability that the other team in the final would be Uruguay (0.7) or the Netherlands (0.3). Thus, the probability that the final match is between Germany and Uruguay is: 0.6 Ã— 0.7 = 0.42. And the probability that the final match is between Germany and the Netherlands is: 0.6 Ã— 0.3 = 0.18. (These add up to 0.6, which makes sense, because that is the probability that Germany makes it to the final.)
Now, we have to determine the probability that Germany wins each of those to possible games — we know that the probability that Uruguay wins is 0.7, so the probability that Germany would win is 0.3. That means the probability that the Germany vs. Uruguay final game takes place AND that Germany wins is 0.42 Ã— 0.3 = 0.126. And we know the probability that Germany would beat the Netherlands, a European team, is 0.6, so the probability that the Germany vs. the Netherlands game takes place AND that Germany wins is 0.18 Ã— 0.6 = 0.108.
By adding those two probabilities together, we cover all the possible outcomes in which Germany takes home the cup. So, the probability that Germany wins the final is 0.126 + 0.108 = 0.234. Since we know this from Statement 1 alone, the statement is Sufficient.
Now we look at Statement 2 alone. It tells us the probability that the Netherlands would win in the final match, were they to make it that far, is 0.4. But this tells us nothing about Germany — maybe the probability Germany makes it to the final match is 0 (sorry, fans). In that case, they certainly are not winning the whole thing; the probability of winning the final is thus 0, too. But maybe Germany has a great chance of making it. We just do not know from the information we have, so Statement 2 is insufficient.
(This would be answer choice A.)