At the mere mention of probability on the GRE students’ eyes glaze over and all conversation stops, save perhaps for the hushed admission that they don’t like probability.
While probability can be very difficult—and many of the publishers give it short shrift (so there are a lack of practice problems)—keeping a few concepts in mind will make this tricky concept seem somewhat less daunting.
First off we need to know the difference between dependent and independent probability. Today, let’s focus on independent probability.
If two events have nothing to do with each other then they are independent (makes sense.) For instance, if I toss a fair-sided coin (on average it will come out heads 50% of the time) twice, then the result of one toss does not affect the other toss. So if I get heads on the first toss, then there is still a 50% chance that I will get heads on the next toss.
To illustrate how this concept isn’t always treated rationally, imagine that you have tossed 8 heads in a row. What is the probability that you will toss heads on the 9th toss? It seems very unlikely that you will toss yet another heads. But again, the 9th toss has nothing to do with what happened on the first eight tosses—the chance of tossing a heads is still 50%. Still, most of us would gamble against heads happening again.
To find the probability of a series independent events happening, we find the probability of each event happening and then multiply those probabilities together.
So let’s go back to our streak of 8 heads in a row. To toss this we will have to multiply the probability of getting heads on one toss (50%) eight times. Let’s convert 50% to ½, for ease of calculation. We get (1/2)^8 =1/256 (not a very likely scenario.)