Mastering Probability and Conditional Probability for the SAT

Welcome to our comprehensive guide on probability and conditional probability for the SAT! These concepts form a crucial part of the Problem-Solving and Data Analysis section of the SAT Math test. Understanding probability will not only help you succeed on the SAT but also develop critical thinking skills that are valuable in many real-world situations. This lesson is designed for students with no prior knowledge of probability, taking you from the basic concepts to solving complex SAT-level problems step by step.

Probability is a mathematical measure of the likelihood that an event will occur. It is expressed as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty.

Basic Probability Formula:

Probability of an event =

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

For example, the probability of rolling a 3 on a fair six-sided die is $\frac{1}{6}$ because there is 1 favorable outcome (rolling a 3) out of 6 possible outcomes.

Conditional Probability is the probability of an event occurring given that another event has already occurred. It's denoted as $P(A|B)$, which reads as "the probability of event A given that event B has occurred."

Conditional Probability Formula:

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

Where:

• $P(A|B)$ is the conditional probability of A given B
• $P(A\cap B)$ is the probability of both A and B occurring (intersection)
• $P(B)$ is the probability of B occurring

Conditional probability helps us update our beliefs about the likelihood of events based on new information.

Step 1: Identify the Sample Space

The sample space is the set of all possible outcomes of an experiment. For example, when flipping a coin, the sample space is {Heads, Tails}.

Step 2: Determine Favorable Outcomes

Identify the outcomes that satisfy the condition you're looking for. For example, if you're calculating the probability of drawing a king from a standard deck of cards, the favorable outcomes are the 4 kings in the deck.

Step 3: Calculate Basic Probability

Use the formula: $P(\text{Event})=\frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$

For example, the probability of drawing a king from a standard 52-card deck is $P(\text{King})=\frac{4}{52}=\frac{1}{13}$

Step 4: For Conditional Probability Problems

1. Identify the condition (the given event B)
2. Determine the new sample space (only outcomes where B occurs)
3. Calculate the probability of event A within this new sample space

Step 5: Use the Conditional Probability Formula When Needed

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

Additional Probability Rules:

Addition Rule (for mutually exclusive events):
$P(A\text{ or }B)=P(A)+P(B)$

Addition Rule (for non-mutually exclusive events):
$P(A\text{ or }B)=P(A)+P(B)-P(A\text{ and }B)$

Multiplication Rule (for independent events):
$P(A\text{ and }B)=P(A)\times P(B)$

Multiplication Rule (for dependent events):
$P(A\text{ and }B)=P(A)\times P(B|A)$

SAT-Specific Tips:

• Draw diagrams or tables to organize information
• Use fractions rather than decimals for clearer work
• Check if events are independent or dependent
• Look for keywords like "given," "if," or "knowing that" which signal conditional probability

Basic Probability Practice

1. A fair six-sided die is rolled. Find the probability of rolling:
   a) A number greater than 4
   b) An even number
   c) A number less than 7

2. A bag contains 5 red marbles, 3 blue marbles, and 2 green marbles. If a marble is drawn at random, find the probability of drawing:
   a) A red marble
   b) A blue or green marble
   c) A yellow marble

Conditional Probability Practice

3. A standard deck of 52 cards is shuffled. Find the probability of:
   a) Drawing a king
   b) Drawing a king given that the card is a face card
   c) Drawing a heart given that the card is red

4. In a class of 30 students, 18 are taking math, 15 are taking science, and 10 are taking both. If a student is selected at random, find the probability that:
   a) The student is taking math given that they are taking science
   b) The student is taking science given that they are taking math

Combined Probability Practice

5. A survey found that 65% of people like chocolate ice cream, 45% like vanilla ice cream, and 25% like both. Find the probability that a randomly selected person:
   a) Likes either chocolate or vanilla ice cream
   b) Likes chocolate ice cream given that they like vanilla ice cream
   c) Likes neither chocolate nor vanilla ice cream

6. Two fair dice are rolled. Find the probability that:
   a) The sum is 7
   b) The sum is 7 given that at least one die shows a 4
   c) Both dice show the same number

Solutions are provided at the end of this worksheet.

1. Confusing Independent and Dependent Events

Many students incorrectly assume that all events are independent. In reality, many real-world scenarios involve dependent events, where the occurrence of one event affects the probability of another event.

Example: When drawing cards without replacement, each draw changes the composition of the deck, making the events dependent.

2. The Gambler's Fallacy

This is the belief that if an event has occurred more frequently than normal in the past, it's less likely to happen in the future (or vice versa).

Example: Thinking that after flipping 5 heads in a row, a tail is "due" or more likely on the next flip. In reality, each flip of a fair coin has a 50% chance of being heads, regardless of previous flips.

3. Misunderstanding Conditional Probability

Students often confuse P(A|B) with P(B|A), which are generally not equal.

Example: The probability of having a disease given a positive test result P(Disease|Positive) is not the same as the probability of a positive test result given that you have the disease P(Positive|Disease).

4. Addition Rule Errors

Incorrectly adding probabilities without accounting for overlap between events.

Example: If P(A) = 0.5 and P(B) = 0.3, the probability of either A or B occurring is not simply 0.5 + 0.3 = 0.8 if the events can occur simultaneously. You must subtract the probability of both occurring: P(A or B) = P(A) + P(B) - P(A and B).

5. Sample Space Confusion

Not correctly identifying the total number of possible outcomes, especially in complex scenarios.

Example: When calculating the probability of getting a sum of 7 when rolling two dice, some students count only the 6 possible outcomes for each die rather than the 36 possible combinations.

6. Overlooking the "At Least One" Scenario

Struggling to calculate the probability of "at least one" occurrence of an event.

Example: To find the probability of getting at least one 6 when rolling three dice, it's often easier to calculate 1 - P(no 6's) rather than adding P(exactly one 6) + P(exactly two 6's) + P(exactly three 6's).