Probability and conditional probability

Probability measures the likelihood of an event occurring, expressed as a number between $0$ (impossible) and $1$ (certain), or as a percentage from $0\%$ to $100\%$.

Basic Probability Principles

The probability of an event $A$, written as $P(A)$, is calculated by comparing desired outcomes to the total possible outcomes.

• Basic Probability Formula:

  $$P(A) = \dfrac{\text{Number of Favorable Outcomes}}{\text{Total Number of Possible Outcomes}}$$

  Example: Rolling a $4$ on a standard six-sided die $\implies P(\text{rolling } 4) = \frac{1}{6}$.

• Complement Rule: The probability that an event does not happen is $1$ minus the probability that it does happen.

  $$P(\text{not } A) = 1 - P(A)$$

  Example: If the probability of rain is $0.3$, the probability of no rain is $1 - 0.3 = 0.7$.

• Mutually Exclusive Events: Two events that cannot happen at the same time.

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

  Example: Rolling a $2$ or a $5$ on a single die roll $\implies \frac{1}{6} + \frac{1}{6} = \frac{2}{6} = \frac{1}{3}$.

Conditional Probability and Two-Way Tables

Conditional probability measures the likelihood of an event occurring given that another condition has already happened.

• Conditional Probability Formula: The notation $P(A \mid B)$ reads "the probability of $A$ given $B$."

  $$P(A \mid B) = \dfrac{P(A \text{ and } B)}{P(B)} = \dfrac{\text{Number of outcomes in both } A \text{ and } B}{\text{Total number of outcomes in condition } B}$$

• Two-Way Tables: SAT probability questions frequently present data in two-way tables. To find conditional probability, restrict your sample space to only the row or column specified by the condition.

  Example: Consider the table below:

  	Passed	Failed	Total
  Studied	$40$	$5$	$45$
  Did Not Study	$10$	$15$	$25$
  Total	$50$	$20$	$70$

  • Probability of selecting someone who passed given that they studied:

    • Condition: Look only at the "Studied" row (total $= 45$).

    • Calculation: $\frac{40}{45} = \frac{8}{9}$.

1. Use row/column totals for conditional probability versus overall totals for general probability.

   Example: Based on the study group data table below:

   	Passed Exam	Failed Exam	Total
   Attended Review Session	$32$	$4$	$36$
   Did Not Attend	$18$	$16$	$34$
   Total	$50$	$20$	$70$

   • General probability (selected at random from all students):

     • Fraction who attended review and passed: $\frac{32}{70} = \frac{16}{35}$

   • Conditional probability (given they attended):

     • Fraction who passed out of attendees: $\frac{32}{36} = \frac{8}{9}$

2. Solve "at least one" problems using the complement rule to save computation steps.

   Example: Based on the vehicle inventory table below:

   	Sedan	SUV	Total
   New	$15$	$25$	$40$
   Used	$30$	$10$	$40$
   Total	$45$	$35$	$80$

   • Question: What fraction of the SUVs are new?

     • "Given" group: All SUVs (Column total $= 35$)

     • Target: New SUVs ($25$)

     • Probability: $\frac{25}{35} = \frac{5}{7}$

3. Determine if sequential events require multiplying probabilities.

   Example: Based on the voter survey table below:

   	In Favor	Opposed	Total
   Group A	$40$	$10$	$50$
   Group B	$20$	$30$	$50$
   Total	$60$	$40$	$100$

   • Question: Among the voters who opposed, what fraction belonged to Group B?

     • Condition phrase: "Among the voters who opposed" (Opposed column total $= 40$)

     • Target: Group B ($30$)

     • Probability: $\frac{30}{40} = \frac{3}{4}$