#AP Statistics: Probability Review 🚀

Hey there, future AP Stats superstar! Let's get you prepped and feeling confident for your exam. This guide is designed to be your go-to resource for a last-minute review, focusing on clarity, key concepts, and practical tips. Let's dive in!

#Visualizing Probability

Probability isn't just about numbers; it's about understanding how likely events are. Here's a quick rundown of how we visualize probabilities:

• Probability Histograms: Bar graphs showing the probability of different outcomes. Think of them as a visual way to see which outcomes are more likely than others. The x-axis shows outcomes, and the y-axis shows probabilities.
• Plots: Graphs showing the relationship between the probability of an event and a continuous variable. Useful for understanding trends and patterns.

Tree Diagrams:

Diagrams showing outcomes in a series of events. Each branch represents a different outcome, and the branch size can represent probability. 🌲 - Great for multi-stage processes. - Probabilities on branches are often conditional.

![Tree Diagram](https://zupay.blob.core.windows.net/resources/files/0baca4f69800419293b4c75aa2870acd_853769_1491.png?alt=media&token=ee32fe54-7711-438f-bdd7-4abd508a1f7e)

*Example of a Tree Diagram*

-
Venn Diagrams:

Diagrams showing relationships between sets of outcomes. The area of each circle represents the probability of that set. 🔵 - Helpful for visualizing intersections (AND) and unions (OR).

![Venn Diagram](https://zupay.blob.core.windows.net/resources/files/0baca4f69800419293b4c75aa2870acd_fd2565_1548.png?alt=media&token=212b2ef8-9019-406d-baed-6a115a0737c8)

*Example of a Venn Diagram*

• Two-Way Tables: Tables that organize data into rows and columns, showing the frequency of different outcomes.

  • Excellent for calculating joint and conditional probabilities.

  

  Example of a Two-Way Table

#Conditional Probability: The "Given" Game

Conditional probability is all about finding the probability of an event given that another event has already occurred. It's written as P(B | A), which means "the probability of B given A." The key word here is "GIVEN."

#The Formula

The formula is:

P(B | A) = P(A ∩ B) / P(A)

Or, in plain English:

P(B given A) = P(A and B) / P(A)

#General Multiplication Rule

We can rearrange the formula to get the general multiplication rule:

P(A and B) = P(A) * P(B | A) 📜

#Practice Problems

Let's put these concepts into action with some practice problems!

#Practice Problem 1: Gene Therapy 🧬

A biotech company is testing a new gene therapy. Here's the data:

• 100 patients in the trial.
• 75 responded to therapy.
• 25 did not respond.
• 50 experienced side effects.
• 50 did not experience side effects.

We want to find:

(A) P(response | side effects) (B) P(no response | no side effects)

#Solution

(A) P(response | side effects) = P(response and side effects) / P(side effects)

• P(response) = 75/100 = 0.75
• P(side effects) = 50/100 = 0.5

• We're missing P(response and side effects), but we can use the general multiplication rule to find it: P(response and side effects) = P(response) * P(side effects | response). However, we don't know P(side effects | response) yet. Let's use the formula P(response | side effects) = P(response and side effects) / P(side effects) and rearrange it: P(response and side effects) = P(response | side effects) * P(side effects). We still don't know P(response | side effects). Let's use the general multiplication rule to find P(response and side effects) = P(response) * P(side effects | response). We are still missing P(side effects | response) and P(response and side effects). Let's assume that the probability of response and side effects is 0.375 (as calculated in the original document).

• P(response | side effects) = 0.375 / 0.5 = 0.75

There's a 75% probability that a patient will respond to the therapy given they experienced side effects.

(B) P(no response | no side effects) = P(no response and no side effects) / P(no side effects)

• P(no response) = 25/100 = 0.25

• P(no side effects) = 50/100 = 0.5

• Let's assume that the probability of no response and no side effects is 0.125 (as calculated in the original document).

• P(no response | no side effects) = 0.125 / 0.5 = 0.25

There's a 25% probability that a patient will not respond to the therapy given they did not experience side effects.

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