Mastering Probability Questions on the SAT

On the Digital SAT exam, probability questions ask you to determine how likely a particular event is to occur. For example, how likely is it that you’ll pick a red marble out of a bag? How likely is it that a particular person will be chosen in a lottery?

This guide will take you through all the aspects of probability you’ll need to know for the SAT, including what probability means, typical probability questions you’ll see on the SAT math section, and the steps needed to solve them.

With 2 math section modules and 44 questions in total, there's a very good chance that at least one probability question will show up. However, it should be noted that there usually aren't more than one or two of these questions per exam, so prioritize your studying accordingly.

Step 1: Understand the Basics of Probability

Probability is the measure of how likely an event is to occur. It is expressed as a fraction where the numerator is the number of desired outcomes and the denominator is the total number of possible outcomes.

For example, the probability of getting tails when flipping a coin is $\frac{1}{2}$ because there is 1 desired outcome (tails) and 2 possible outcomes (heads and tails).

Simple Probability

This involves finding the probability of a single event occurring.

Example: The probability of drawing a red marble from a bag with 5 red and 5 blue marbles is $\frac{5}{10} = \frac{1}{2}$.

Step-by-step:

1. Identify the total number of possible outcomes. Here, there are 10 marbles in total (5 red and 5 blue).
2. Identify the number of desired outcomes. The desired outcome is drawing a red marble, of which there are 5.
3. Set up the probability fraction as the number of desired outcomes over the total number of possible outcomes: $\frac{5}{10}$.
4. Simplify the fraction by dividing the numerator and the denominator by their greatest common divisor (5): $\frac{5}{10} = \frac{1}{2}$.

Compound Probability

This involves finding the probability of two or more events occurring together.

Example: The probability of drawing two red marbles in a row from the same bag (without replacement) is $\frac{5}{10} \times \frac{4}{9} = \frac{2}{9}$.

Step-by-step:

1. For the first draw, identify the total number of possible outcomes (10 marbles) and the number of desired outcomes (5 red marbles). The probability is $\frac{5}{10}$.
2. For the second draw, since we are not replacing the first marble, there are now 9 marbles left in the bag, with 4 being red. The probability is $\frac{4}{9}$.
3. Multiply the probabilities of the two events to find the compound probability: $\frac{5}{10} \times \frac{4}{9}$.
4. Simplify the fraction: $\frac{5}{10} \times \frac{4}{9} = \frac{2}{9}$.

Either/Or Probability

This involves finding the probability of either one event or another event occurring.

Example: The probability of drawing either an ace or a queen from a deck of cards is $\frac{4}{52} + \frac{4}{52} = \frac{2}{13}$.

Step-by-step:

1. Identify the total number of possible outcomes (52 cards in a deck).
2. Identify the number of desired outcomes for each event. There are 4 aces and 4 queens.
3. Calculate the probability for each event: $\frac{4}{52}$ for drawing an ace and $\frac{4}{52}$ for drawing a queen.
4. Add the probabilities of the two events: $\frac{4}{52} + \frac{4}{52}$.
5. Simplify the fraction: $\frac{8}{52} = \frac{2}{13}$.

Conditional Probability

Conditional probability is the chance of an event (B) happening given that another event or condition (A) has already happened. It still uses the basic formula of desired outcomes over total outcomes, but identifying these outcomes can be trickier.

Example: If there are 100 people working on a performance, including 52 dancers, and among the dancers, 14 are ballet dancers, the probability of selecting a ballet dancer given that the person selected is a dancer is $\frac{14}{52}$.

Step-by-step:

1. Identify the total number of outcomes based on the given condition. Here, the condition is that the person selected is a dancer, so there are 52 dancers.
2. Identify the number of desired outcomes within this group. The desired outcome is selecting a ballet dancer, of which there are 14.
3. Set up the probability fraction as the number of desired outcomes over the total number of possible outcomes given the condition: $\frac{14}{52}$.

Step 2: Identify the Type of Probability Question

It's important to determine whether the question is asking for simple probability, compound probability, either/or probability, or conditional probability.

Simple Probability

Look for questions asking for the likelihood of a single event occurring.

Example: What is the probability of rolling a 4 on a six-sided die? $\frac{1}{6}$.

Compound Probability

Look for questions asking for the likelihood of multiple events occurring together.

Example: What is the probability of rolling two 4s in a row on a six-sided die? $\frac{1}{6} \times \frac{1}{6} = \frac{1}{36}$.

Either/Or Probability

Look for questions asking for the likelihood of one event or another occurring.

Example: What is the probability of drawing either a red or blue marble from a bag with 5 red and 5 blue marbles? $\frac{5}{10} + \frac{5}{10} = 1$.

Conditional Probability

Look for questions indicating a precondition, such as "given" or "assuming." These questions ask for the probability of an event occurring given that another event has already occurred.

Example: If 52 dancers, 12 stage technicians, and 36 musicians are working on a performance, and we need to find the probability of selecting a ballet dancer given the person selected is a dancer, it would be $\frac{14}{52}$.

Step 3: Apply the Appropriate Formula

Once you have identified the type of probability question, apply the corresponding formula to solve the problem.

Simple Probability Formula

Simple probability refers to the likelihood of a single event occurring. It is calculated using the ratio of the number of favorable outcomes to the total number of possible outcomes.

The formula for simple probability is:

$P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$ where:

• $P(E)$ is the probability of event E occurring.
• The numerator represents the number of favorable outcomes for the event.
• The denominator represents the total number of possible outcomes in the sample space.

To solve a simple probability question, follow these steps:

1. Identify the Total Number of Possible Outcomes: Determine the total number of possible outcomes for the event.
2. Determine the Number of Favorable Outcomes: Identify the number of outcomes that favor the event you are calculating the probability for.
3. Apply the Formula: Use the simple probability formula to calculate the probability of the event.

Example: What is the probability of rolling a 4 on a six-sided die?

Step 1: Identify the Total Number of Possible Outcomes: A six-sided die has 6 possible outcomes (1, 2, 3, 4, 5, 6).

Step 2: Determine the Number of Favorable Outcomes: There is only 1 favorable outcome (rolling a 4).

Step 3: Apply the Formula: Use the formula $P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$:

$P(\text{rolling a 4}) = \frac{1}{6}$.

Therefore, the probability of rolling a 4 on a six-sided die is $\frac{1}{6}$.

Another example: What is the probability of drawing a red marble from a bag containing 5 red marbles, 3 green marbles, and 2 blue marbles?

Step 1: Identify the Total Number of Possible Outcomes: The bag contains 10 marbles in total (5 red, 3 green, 2 blue).

Step 2: Determine the Number of Favorable Outcomes: There are 5 favorable outcomes (drawing a red marble).

Step 3: Apply the Formula: Use the formula $P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$:

$P(\text{drawing a red marble}) = \frac{5}{10} = \frac{1}{2}$.

Therefore, the probability of drawing a red marble from the bag is $\frac{1}{2}$.

Another example: What is the probability of selecting a boy from a class of 20 students where 12 are girls and 8 are boys?

Step 1: Identify the Total Number of Possible Outcomes: The class contains 20 students in total.

Step 2: Determine the Number of Favorable Outcomes: There are 8 favorable outcomes (selecting a boy).

Step 3: Apply the Formula: Use the formula $P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$:

$P(\text{selecting a boy}) = \frac{8}{20} = \frac{2}{5}$.

Therefore, the probability of selecting a boy from the class is $\frac{2}{5}$.

Compound Probability Formula

Compound probability refers to the likelihood of two or more events occurring together. It can be calculated for both dependent and independent events.

The formula for compound probability of independent events is:

$P(A \text{ and } B) = P(A) \times P(B)$ where:

• $P(A)$ is the probability of event A occurring.
• $P(B)$ is the probability of event B occurring.
• The multiplication rule is used because the occurrence of event A does not affect the occurrence of event B.

The formula for compound probability of dependent events is:

$P(A \text{ and } B) = P(A) \times P(B|A)$ where:

• $P(A)$ is the probability of event A occurring.
• $P(B|A)$ is the probability of event B occurring given that event A has already occurred.
• The conditional probability is used because the occurrence of event A affects the probability of event B occurring.

To solve a compound probability question, follow these steps:

1. Determine if the Events are Independent or Dependent: Identify whether the events affect each other.
2. Calculate the Probability of Each Event: Find the probability of each individual event.
3. Apply the Appropriate Formula: Use the correct formula based on whether the events are independent or dependent.

Example (Independent Events): What is the probability of rolling a 4 on a six-sided die and flipping heads on a coin?

Step 1: Determine if the Events are Independent or Dependent: Rolling a die and flipping a coin are independent events.

Step 2: Calculate the Probability of Each Event: The probability of rolling a 4 is $\frac{1}{6}$ and the probability of flipping heads is $\frac{1}{2}$.

Step 3: Apply the Appropriate Formula: Use the formula $P(A \text{ and } B) = P(A) \times P(B)$:

$P(\text{rolling a 4 and flipping heads}) = \frac{1}{6} \times \frac{1}{2} = \frac{1}{12}$.

Therefore, the probability of rolling a 4 and flipping heads is $\frac{1}{12}$.

Example (Dependent Events): What is the probability of drawing two aces in a row from a deck of cards without replacement?

Step 1: Determine if the Events are Independent or Dependent: Drawing cards without replacement are dependent events.

Step 2: Calculate the Probability of Each Event: The probability of drawing the first ace is $\frac{4}{52}$. After drawing the first ace, the probability of drawing the second ace is $\frac{3}{51}$.

Step 3: Apply the Appropriate Formula: Use the formula $P(A \text{ and } B) = P(A) \times P(B|A)$:

$P(\text{drawing two aces}) = \frac{4}{52} \times \frac{3}{51} = \frac{1}{221}$.

Therefore, the probability of drawing two aces in a row without replacement is $\frac{1}{221}$.

Another example (Independent Events): What is the probability of flipping two heads in a row with a fair coin?

Step 1: Determine if the Events are Independent or Dependent: Flipping a coin multiple times are independent events.

Step 2: Calculate the Probability of Each Event: The probability of flipping heads each time is $\frac{1}{2}$.

Step 3: Apply the Appropriate Formula: Use the formula $P(A \text{ and } B) = P(A) \times P(B)$:

$P(\text{flipping two heads}) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$.

Therefore, the probability of flipping two heads in a row is $\frac{1}{4}$.

Another example (Dependent Events): What is the probability of selecting two red marbles in a row from a bag containing 5 red marbles and 5 blue marbles without replacement?

Step 1: Determine if the Events are Independent or Dependent: Selecting marbles without replacement are dependent events.

Step 2: Calculate the Probability of Each Event: The probability of selecting the first red marble is $\frac{5}{10}$. After selecting the first red marble, the probability of selecting the second red marble is $\frac{4}{9}$.

Step 3: Apply the Appropriate Formula: Use the formula $P(A \text{ and } B) = P(A) \times P(B|A)$:

$P(\text{selecting two red marbles}) = \frac{5}{10} \times \frac{4}{9} = \frac{2}{9}$.

Therefore, the probability of selecting two red marbles in a row without replacement is $\frac{2}{9}$.

Either/Or Probability Formula

Either/or probability, also known as the probability of the union of two events, refers to the likelihood of either one event or another event occurring. This can be calculated for both mutually exclusive and non-mutually exclusive events.

The formula for either/or probability for mutually exclusive events (events that cannot happen at the same time) is:

$P(A \text{ or } B) = P(A) + P(B)$ where:

• $P(A)$ is the probability of event A occurring.
• $P(B)$ is the probability of event B occurring.
• Since the events are mutually exclusive, there is no overlap between them.

The formula for either/or probability for non-mutually exclusive events (events that can happen at the same time) is:

$P(A \text{ or } B) = P(A) + P(B) - P(A \cap B)$ where:

• $P(A)$ is the probability of event A occurring.
• $P(B)$ is the probability of event B occurring.
• $P(A \cap B)$ is the probability of both events A and B occurring together (the overlap).

To solve an either/or probability question, follow these steps:

1. Determine if the Events are Mutually Exclusive or Non-Mutually Exclusive: Identify whether the events can happen at the same time.
2. Calculate the Probability of Each Event: Find the probability of each individual event.
3. Apply the Appropriate Formula: Use the correct formula based on whether the events are mutually exclusive or non-mutually exclusive.

Example (Mutually Exclusive Events): What is the probability of drawing either an ace or a king from a deck of cards?

Step 1: Determine if the Events are Mutually Exclusive or Non-Mutually Exclusive: Drawing an ace and drawing a king are mutually exclusive events because one card cannot be both an ace and a king.

Step 2: Calculate the Probability of Each Event: The probability of drawing an ace is $\frac{4}{52}$ and the probability of drawing a king is $\frac{4}{52}$.

Step 3: Apply the Appropriate Formula: Use the formula $P(A \text{ or } B) = P(A) + P(B)$:

$P(\text{drawing an ace or a king}) = \frac{4}{52} + \frac{4}{52} = \frac{8}{52} = \frac{2}{13}$.

Therefore, the probability of drawing either an ace or a king is $\frac{2}{13}$.

Example (Non-Mutually Exclusive Events): What is the probability of drawing a heart or a face card from a deck of cards?

Step 1: Determine if the Events are Mutually Exclusive or Non-Mutually Exclusive: Drawing a heart and drawing a face card are non-mutually exclusive events because a card can be both a heart and a face card (e.g., the king of hearts).

Step 2: Calculate the Probability of Each Event: The probability of drawing a heart is $\frac{13}{52}$ and the probability of drawing a face card is $\frac{12}{52}$. The probability of drawing a card that is both a heart and a face card (king, queen, or jack of hearts) is $\frac{3}{52}$.

Step 3: Apply the Appropriate Formula: Use the formula $P(A \text{ or } B) = P(A) + P(B) - P(A \cap B)$:

$P(\text{drawing a heart or a face card}) = \frac{13}{52} + \frac{12}{52} - \frac{3}{52} = \frac{22}{52} = \frac{11}{26}$.

Therefore, the probability of drawing either a heart or a face card is $\frac{11}{26}$.

Another example (Mutually Exclusive Events): What is the probability of rolling either a 2 or a 5 on a six-sided die?

Step 1: Determine if the Events are Mutually Exclusive or Non-Mutually Exclusive: Rolling a 2 and rolling a 5 are mutually exclusive events because a single roll cannot result in both a 2 and a 5.

Step 2: Calculate the Probability of Each Event: The probability of rolling a 2 is $\frac{1}{6}$ and the probability of rolling a 5 is $\frac{1}{6}$.

Step 3: Apply the Appropriate Formula: Use the formula $P(A \text{ or } B) = P(A) + P(B)$:

$P(\text{rolling a 2 or a 5}) = \frac{1}{6} + \frac{1}{6} = \frac{2}{6} = \frac{1}{3}$.

Therefore, the probability of rolling either a 2 or a 5 is $\frac{1}{3}$.

Another example (Non-Mutually Exclusive Events): What is the probability of selecting a student who is either a senior or on the soccer team, if there are 40 students in a class, 15 are seniors, 10 are on the soccer team, and 5 are seniors on the soccer team?

Step 1: Determine if the Events are Mutually Exclusive or Non-Mutually Exclusive: Being a senior and being on the soccer team are non-mutually exclusive events because a student can be both a senior and on the soccer team.

Step 2: Calculate the Probability of Each Event: The probability of selecting a senior is $\frac{15}{40}$ and the probability of selecting a student on the soccer team is $\frac{10}{40}$. The probability of selecting a senior on the soccer team is $\frac{5}{40}$.

Step 3: Apply the Appropriate Formula: Use the formula $P(A \text{ or } B) = P(A) + P(B) - P(A \cap B)$:

$P(\text{selecting a senior or a soccer player}) = \frac{15}{40} + \frac{10}{40} - \frac{5}{40} = \frac{20}{40} = \frac{1}{2}$.

Therefore, the probability of selecting a student who is either a senior or on the soccer team is $\frac{1}{2}$.

Conditional Probability Formula

Conditional probability is calculated by focusing only on the subset of outcomes that meet the given condition.

The formula for conditional probability is:

$P(B|A) = \frac{P(A \cap B)}{P(A)}$ where:

• $P(B|A)$ is the probability of event B occurring given that event A has occurred.
• $P(A \cap B)$ is the probability of both events A and B occurring together.
• $P(A)$ is the probability of event A occurring.

To solve a conditional probability question, follow these steps:

1. Identify the Given Condition: Determine which event is given (event A) and which event's probability you need to find given this condition (event B).
2. Calculate the Probability of Both Events Occurring: Find the probability of both event A and event B happening together.
3. Calculate the Probability of the Given Condition: Determine the probability of event A occurring.
4. Apply the Formula: Use the conditional probability formula to find the probability of event B given event A.

Example: There are 100 people working on a performance: 52 dancers, 12 stage technicians, and 36 musicians. Among the dancers, 14 are ballet dancers. What is the probability of selecting a ballet dancer given that the person selected is a dancer?

Step 1: Identify the Given Condition: The given condition is that the person selected is a dancer (event A).

Step 2: Calculate the Probability of Both Events Occurring: The probability of selecting a ballet dancer (event B) and the person being a dancer is $\frac{14}{100}$ because there are 14 ballet dancers out of 100 people.

Step 3: Calculate the Probability of the Given Condition: The probability of selecting a dancer (event A) is $\frac{52}{100}$ because there are 52 dancers out of 100 people.

Step 4: Apply the Formula: Use the formula $P(B|A) = \frac{P(A \cap B)}{P(A)}$ to find the conditional probability:

$P(\text{ballet dancer}|\text{dancer}) = \frac{\frac{14}{100}}{\frac{52}{100}} = \frac{14}{52} = \frac{7}{26}$.

Therefore, the probability of selecting a ballet dancer given that the person selected is a dancer is $\frac{7}{26}$.

Another example: What is the probability of selecting a red marble from a bag given that the marble is not blue, if the bag contains 5 red marbles, 3 green marbles, and 2 blue marbles?

Step 1: Identify the Given Condition: The given condition is that the marble selected is not blue (event A).

Step 2: Calculate the Probability of Both Events Occurring: The probability of selecting a red marble (event B) and the marble not being blue is $\frac{5}{10}$ because there are 5 red marbles out of 10 total marbles.

Step 3: Calculate the Probability of the Given Condition: The probability of selecting a marble that is not blue (event A) is $\frac{8}{10}$ because there are 8 marbles that are not blue out of 10 total marbles.

Step 4: Apply the Formula: Use the formula $P(B|A) = \frac{P(A \cap B)}{P(A)}$ to find the conditional probability:

$P(\text{red marble}|\text{not blue}) = \frac{\frac{5}{10}}{\frac{8}{10}} = \frac{5}{8}$.

Therefore, the probability of selecting a red marble given that the marble is not blue is $\frac{5}{8}$.

Step 4: Verify Your Answer

Check your work to ensure your answer makes sense in the context of the problem. Re-read the question and confirm that you have identified the correct type of probability and applied the correct formula.

Complete Example

Let's go through a typical SAT probability question:

Given a table of students who recalled dreams, what is the probability of selecting a student from Group Y who recalled at least one dream?

Group X: 28 students recalled 1-4 dreams, 57 recalled 5+ dreams.

Group Y: 11 students recalled 1-4 dreams, 68 recalled 5+ dreams.

Total: 164 students recalled at least one dream.

Number of desired outcomes (students from Group Y): $11 + 68 = 79$.

Probability: $\frac{79}{164}$.

Another Example

Consider a deck of cards. What is the probability of drawing either an ace or a queen?

There are 4 aces and 4 queens in a deck of 52 cards. So, the probability of drawing an ace is \frac{4}{52} and the probability of drawing a queen is $\frac{4}{52}$.

Therefore, the probability of drawing either an ace or a queen is $\frac{4}{52} + \frac{4}{52} = \frac{8}{52} = \frac{2}{13}$.

Conditional Probability Example

What is the probability of selecting a ballet dancer given that the person selected is a dancer from a group of performers including 52 dancers?

Among the 52 dancers, 14 are ballet dancers. Therefore, the probability is $\frac{14}{52}$.