Summary of the Lecture on Probability Using a Tree Diagram
The class focused on solving a probability problem using a tree diagram. The problem involved Jenny picking sweets from a bag containing both blue and red sweets, with replacement between picks. The professor aimed to calculate various probabilities using the tree diagram and emphasized the concepts of multiplying probabilities along the branches and adding results for inclusive outcomes.
Detailed Notes on the Probability Problem
Problem Statement
- Jenny has a bag with 7 blue sweets and 3 red sweets.
- She picks a sweet at random, replaces it, and then picks another sweet at random.
Creating a Tree Diagram
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Initial Assumptions:
- Total sweets = 10 (7 blue + 3 red).
- Probabilities for the first pick:
- Blue (B) = 7/10
- Red (R) = 3/10
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Second Pick (Same as First Due to Replacement):
- Probabilities remain the same because of replacement.
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Tree Diagram Structure:
- First level: First pick (B or R)
- Second level: Second pick following each first pick outcome (B or R)
- Branch Probabilities: Multiply probabilities along branches.
Calculating Outcomes and Probabilities
- BB (Blue then Blue): (7/10) x (7/10) = 49/100
- BR (Blue then Red): (7/10) x (3/10) = 21/100
- RB (Red then Blue): (3/10) x (7/10) = 21/100
- RR (Red then Red): (3/10) x (3/10) = 9/100
- Check: The sum of all probabilities = 1 (100/100)
Specific Probability Questions
- Two red sweets (RR): 9/100
- No red sweets (BB): 49/100
- At least one blue sweet (BR, RB, BB):
- By Addition: 21/100 + 21/100 + 49/100 = 91/100
- Alternative Method: 1 (total probability) - 9/100 (RR) = 91/100
- One sweet of each color (BR or RB):
- By Addition: 21/100 + 21/100 = 42/100
- Simplified to 21/50
Summary of Operations
- Multiplication: Used along the branches for sequential independent events.
- Addition: Used to calculate the total probability of mutually exclusive combined outcomes.
Visual Reminder: Remember to visualize the tree diagram for a clearer understanding of how the probabilities are structured and combined. This visual tool aids in learning how probability works for different independent and combined events.