Probability questions are quite common on the SAT. Probability is the “odds” of any particular event happening. It is expressed as a fraction of: the likelihood of the event over all the outcomes possible.

A very simple real world example is – How likely is it that we will get heads when we flip a coin?

Total possible outcomes are two – either heads or tails.

Number of desired outcomes is one – heads

Hence, probability = ½

Let us take a look at an another example:

There are ten students in the class. Every day, the teacher selects a random student to recite a poem. What are the odds that Student A will be selected to recite the poem today?

Total possible outcomes =10 because there are 10 students to pick from

Desire outcomes = 1 i.e. student A gets selected

The probability of Student A being selected = 1/10

Because the probability of any event happening is expressed as a fraction, it means that an event that will absolutely and without a doubt occur has a probability of or 1. There is no higher chance of it happening–this particular event will happen every single time, without fail.

Probability of an absolutely impossible event, will be 0 because = 0

Probability can also be expressed as a percentage and some probability questions on the SAT will ask you for the answer in percentage form. In the above example of students reciting poem, the probability = 10%

Either/Or Probability

In these kinds of probability questions (called “non-overlapping” probability quetions), the two events cannot both happen at the same time.

So essentially, the number of desired outcomes is increased.

Suppose we change the above example to – There are ten students in the class. Every day, the teacher selects a random student to recite a poem. What are the odds that Student A, B or C will be selected to recite the poem today?

Number of desired outcomes = 3

Total possible outcomes = 10

Probability = 3/10 = 30%

Neither/nor Probability

This is the probability that neither two nor more events will occur. Basically, the desired outcomes are all the potential outcomes minus the ones you highlighted as undesirable. This type of probability again deals with non-overlapping events.

Probability of neither event

Suppose we change our example question to ask – What are the odds that Student A, B or C will not be selected to recite the poem today?

Probability = 1 – 3/10 = 7/10

Combined Probability

Combined probability questions are those in which the probability of two or more events happening together. This is a “both/and” question requires that multiple events all occur.

An “either/or” question requires you to add your probabilities. A “both/and” question requires you to multiply them.

A good rule of thumb is

Probability of multiple events happening will ultimately have a lower probability than the odds of just one of those events happening. Why? Because each events occurring will have probability . When we calculate combined probability, we multiply probability of all events. And when we multiply a number with something , it remains the same or decreases. Therefore, multiple events naturally have lower odds than those of just one event. g. How likely is it that your first and second coin tosses will both be tails? Lower than the odds of just flipping tails once.

An either/or probability will have higher odds than the probability of just one of its events happening. We are adding two probabilities and hence, the odds of getting a desirable outcome will increase. How likely is it that you’ll flip either heads or tails for each toss? 100%! Higher than the odds of flipping a tail

Combined unconditional Events

Unconditional events are independent of one another. In other words, the outcome of one event does not affect the outcome of the second. In this case, when we are asked for the odds of both events occurring, we can simply multiply them. For example,

There are 3 red marbles, 3 green marbles, and 3 blue marbles in a pouch. If Celia draws out a marble and puts it back in, what are the odds that she will draw a red marble AND a blue marble on her first two selections?

The odds of drawing a red marble are 3/9.

Now, she puts the red marble back marble goes to select a blue marble. There are 3 blue marbles out of the 9 marbles in the pouch. The odds of drawing a blue marble are 3/9.

The odds are 1 in 9 (11.11%) that she will select a red marble and a blue marble on her first two selections.

Does this work if she were to select a blue marble first?

The odds of selecting that first blue marble would be 3/9.

The odds of selecting a red marble next would be 3/9.

Therefore, whether she selects red, then blue or blue, then red, the odds remain the same. This is unconditional combined probability.

Conditional Combined Probability

Conditional events affect one another. If two outcomes affect one another, they change the number of possible outcomes. For example,

There are 3 red marbles, 3 green marbles, and 3 blue marbles in a pouch. If Celia draws out marbles without replacing them, what are the odds that she will draw a red marble AND a blue marble on her first two selections?

The odds of drawing a red marble are 3/9.

Now, we are operating under the assumption that Celia managed to select that red marble.

Now, she goes to select a blue marble. There are 3 blue marbles out of the remaining 8 marbles left in the pouch. The odds of drawing a blue marble are 3/8.

The odds are 1 in 8 (12.5%) that she will select a red marble and a blue marble on her first two selections.

Does this work if she were to select a blue marble first?

The odds of selecting that first blue marble would be 3/9.

The odds of selecting a red marble next would be 3/8.

Therefore, whether she selects red, then blue or blue, then red, the odds remain the same. This is conditional combined probability.

Tips for probability questions: