In our previous posts, we've talked about the basic concepts of probability and some fundamental facts about probabilities. Here, we'll show how to calculate the probability of a single event when all the outcome are equally likely. This is, in a sense, the simplest case that we will cover, and it is crucial for everything we'll do later (e.g. in finding the probability of two events occurring).
Suppose we flip a fair coin. What is the probability we get heads? Intuitively, the answer should be 1/2. And that's exactly what the following rule would say:
Probability of Equally Likely Outcomes:
If you have n possible outcomes, all of which are equally likely, then the probability of any particular outcome occurring is 1/n.
So when we flip a fair coin, there are 2 possible outcomes (heads and tails). So n = 2 and the probability of one outcome (e.g. heads) occurring is 1/n = 1/2. And if we roll a six-sided die, there are 6 possible outcomes. So the probability of any particular outcome (e.g. rolling a 4) is 1/6. And if we held a raffle where there were 109 different entrants, the probability of any one of them winning would be 1/109.
Note that this rule only applies when all the outcomes are equally likely. In most GRE problems, the outcomes will be equally likely, and the question will signal that by saying that the outcome is "random" or that the outcomes are "equally likely." So, the question might say things like: "a name is chosen at random" or that "each outcome is equally likely." When the outcomes are not equally likely, all bets are off, and you will have to be more careful in how you approach the problem.
Now, we want to find the probability of some event occurring. Suppose I am going to roll a six-sided die, numbered 1 through 6. What is the probability that I get an even number? To calculate this, we use the following rule:
Probability of Single Events (for equally likely outcomes)
Suppose you have n equally likely outcomes. Then, the probability of some event E occurring is:
where the # of total outcomes = n.
So to find the probability of rolling an even number, we need to find the number of outcomes where we roll an even number. If we roll an even number, then we must have rolled a 2, 4, or 6. Then, we divide by the number of total outcomes, in our case 6. So, P(Roll an even number) = 3/6 = 1/2.
Here’s another example in a similar spirit:
Suppose you randomly choose a number from 1 to 50. What is the probability that you chose a prime number?
There are 50 possible outcomes to your random choice. Now, we need to know: in how many of those outcomes do you choose a prime number? In other words, how many of the numbers 1 to 50 are prime? Here, we just have to go through the list: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47. So there are fifteen such prime numbers. Thus: P(Pick a prime number) = 15/50 = 3/10
Now we know how to find the probability of a single event when the possible outcomes are equally likely. Our next step is to learn how to combine these probabilities in order to get the probabilities of more complex events.
130 people line up to buy raffle tickets. Every 10th person who buys a ticket gets a teddy bear as a promotional item. What is the probability that a randomly chosen person from the line will receive a teddy bear?
There are 130 people in line. Since every 10th person gets a teddy bear, we know 13 people got teddy bears. To find the probability that a randomly selected person gets a teddy bear, we just need to calculate: # of people who get teddy bears/# of people total. Thus, we get 13/130 = 1/10.
You have 50 friends. 12 of them have blue hair. You randomly pick one of your friends to invite to dinner tomorrow. What is the probability that you invite a person with blue hair?
We are looking for P(I invite a person with blue hair). Now, if I randomly pick a friend, that means there are 50 possible outcomes: I get dinner with friend 1, with friend 2, ..., with friend 50. Since the question tells us that the outcome is randomly selected, we know that they are all equally likely. So we can apply our rule. In how many of the outcomes will you get dinner with a blue-haired friend? In 12 of them. And we already know how many total outcomes there are. Thus, P(I invite a person with blue hair) = 12/50 = 6/25.
You still have 50 friends. 12 of them still have blue hair. What is the probability that you do not invite a person with blue hair?
This is just like our previous question, except now we want to find P(I do not invite a person with blue hair). If there are 12 people with blue hair, then there are 50 - 12 = 38 people without blue hair. So there are 38 outcomes where I do not invite a person with blue hair. Again, since the outcomes are chosen randomly, we can apply our principle to get P(I do not invite a person with blue hair) = 38/50 = 19/25.