GMAT Math: Basic Probability
Whether you realise it or not, we are surrounded by probability. When the weather forecast states a 70% chance of rain, that is a probability. When doctors mention our chances of survival, they also use probability. Probability relates to the likelihood of something to happen or not to happen. So everything we do in probability related questions revolves around that simple understanding. It relates to the chance of something happening, but we simply express it in mathematical form. In this post we will have a look at the basic principles of probability, which will help us when we go onto harder problems.
Lets start with an example. Imagine we have a bag that contains five marbles. Three of the marbles are red, and two of the marbles are white.
So we have five marbles of two different colours within the bag.
If you pull a marble from the bag at random, what is the probability that it will be red?
We work out the answer using the formula for simple probability:
'A' in this equation is the event we are trying to calculate. So in this question, A is equal to the number of red marbles within the bag. Re-writing the equation will perhaps make what we are calculating very clear:
In numerical terms, there are 3 red marbles in the bag, and 5 in total, giving us:
Therefore the probability of picking a red marble out of the bag is 3/5 = 0.6 = 60%
On the GMAT you may have to give your answer as a fraction, decimal or percentage depending upon the question, so it's good practice to learn how to convert them quickly. (To turn a decimal probability into a percentage, simply multiply it by 100).
Lets try another example with a very commonly used GMAT tool, a die.
What is the probability of rolling an odd number on a die?
Let us start with the basic probability equation:
In this case 'A' is the number of times an odd number can occur. If we roll a die, there are 6 total possible outcomes as you can of course, roll 1, 2, 3, 4, 5 or 6. That gives us six total possibilities and thus 6 will go on the bottom of our equation.
Out of the six possible outcomes, there are 3 odd numbers: 1, 3 and 5. So the number of times an odd number can occur is 3, and that is the number we put on the top of the equation:
Get used to this basic type of problem.
Adding and Multiplying Probabilities
What we have seen above is the fundamental principle of probability; however, GMAT probability question like to go a few steps further. GMAT will offer questions that will make us think a little harder and will most likely mix two or more probabilities together. In such case, we will need to either multiply probabilities or add them. Here are some general rules to guide you:
If two events have to occur together, generally an "and" is used in such questions. For example, what is the probability of rolling a 5 on a die and then rolling a 2? Both these events must occur together, so in this case we multiply probabilities.
If either of the two events has to occur, then generally an "or" is used. For example, what is the probability of rolling a 5 or a 2 on a die. In this case we can add the probabilities to form an overall probability of either of them occurring.
Let us look at some examples.
What is the probability that you role a 6 on a die and then roll a 2?
We know already that we have to work out two different probabilities here, but now we have to work out whether we add or multiply our answers.
Looking at the wording, we can clearly see that we MUST roll a six first and then roll a 2, so both events must occur, and thus we will multiply the answers.
The probability of rolling a 6, is 1/6
The probability of rolling a 2, is 1/6
The probability of rolling a 6 and then a 2 is:
Another example we can look at is:
There are 500 tickets in a raffle. If Mary buys 2 tickets and John buys 5 tickets, what is the probability that either Mary or John wins the main prize?(Assume that only one person can win the main prize)
In this question we again face two probabilities: the probability of Mary winning and the probability of John winning. However, we need to decide whether we multiply or add the resulting probabilities.
Since either Mary or John can win, it makes sense that we add them.
Mary's chance of winning the jackpot is 2 out of 500 or:
John has 5 tickets, giving him 5 chances out of 500 to win, so his probability is:
Since in our question, either of them can win, it's almost like combining the tickets together so they have 7 tickets out of the 500 tickets or:
So by adding their probabilities together, we can work out the chance that either of them wins.
It is important that you learn to write probabilities as fractions quickly as that is usually the first step in GMAT probability questions. Learn the equation at the top of this post and learn the two rules regarding when you add probabilities and when you multiply. And most importantly—keep practicing!!!
- GMAT Math Section
- GMAT Math:Picking Numbers
- GMAT Math: Backsolving
- GMAT Math: Comparing Fractions
- GMAT Math: Calculation Shortcuts