Probability is one of the most dreaded parts of quant on the GRE. I’ve excerpted the below on what you need to know about it from my book on GRE quant.
How to recognize:
Whenever we talk about the probability or the chance of something happening.
What to know:
1. Our golden rule in probability is always (number of successes)/(total number of options). So, if we have three marbles in a bag that are red, blue, and green respectively, we have a 1/3 chance of picking red.
Because we’re counting the total number of options, this often plays into our counting formulas of permutations and combinations.
Example:
If we’re wondering what the chance is of picking your friends Jeff, Larry, and Bob out of your group of 8 friends if you choose randomly, then:
We first calculate how many ways we can successfully pick Jeff, Larry, and Bob out of 8 people. That’s just 1 way (we need to pick all 3 of them).
Then we can count the total number of ways of picking 3 people out of 8, which is 8C3, or 8!/(3!(8-3)!)=56.
So our probability is 1/56.
2. One of the major factors to consider in probability is whether the probability is independent or dependent. We’ll discuss independent first.
Independent probabilities are like flipping coins. They do not affect each other. Whether one coin gets heads does not affect if the other coin gets heads. If you’re given all the independent probabilities for a given event, they should add up to 1.
Independent probabilities can be related by AND or OR, much like counting, and they do mean the same thing. AND means multiply, OR means add.
We can also add an additional one to that, NOT, which means 1-probability.
Example:
The chance of flipping a coin twice and getting heads AND heads is 1/2*1/2=1/4.
The chance of flipping a coin and getting heads OR tails is 1/2+1/2=1. That makes sense, as those are all the probabilities for the event.
The chance of NOT getting heads is 1-1/2=12. This makes sense, as it’s the same as the probability of getting tails.
3. Dependent probabilities do affect each other, like the chance of it raining and the chance of a given person carrying an umbrella. A given person is much more likely to carry an umbrella if it’s raining.
These are usually best thought of as tossing a dart into a Venn diagram. There’s some overlap between how likely both are to happen which can’t be found just by multiplication.
Example:
If there’s a 40% chance of it raining, a 30% chance of carrying an umbrella, and a 25% chance of both happening, then the chance of neither happening is:
total = A + B – both +neither
100%=40%+30%-25%+neither=55%
Note that in the diagram below, it is pretty unlikely for someone to carry an umbrella without it raining (it only happens 5% of the time).
4. Conditional probabilities are often tested, like what the chance is of something happening and then something else happening.
These can be solved by simply writing out the probabilities for both events, and relating them with AND, OR, or NOT.
Example:
If there are three marbles in a bag, which are red, blue, and green, and you’re picking one marble out at a time and not putting it back, then the chance of picking a red and then a blue is:
1/3*1/2=16. Note that the second number is 1/2 because, if I do not replace the first marble, I only have two total options afterwards.
Takeaways:
1. Golden rule: number of successes over total number of options.
2. Independent probabilities are coin flips, dependent are Venn diagrams.
3. The sum of all mutually exclusive probabilities is 1. 4. AND means multiply, OR means add, NOT means 1-.
If you’re looking for practice with probability along with detailed explanations, you should get my GRE quant book.