Lecture Notes: Probability and Odds

Introduction

• Topic: Exploring probability and defining odds
• Focus: Calculating odds in favor and odds against using probability

Definitions and Concepts

Event and Complement

• Event: The occurrence we are interested in (e.g., winning the gray cup)
• Probability of Event (A):
  • Given as 25% or 0.25 (can also be written as 1/4)
• Complement of Event (A'):
  • The event not occurring
  • Notated as A' or A-bar
  • Probability of Complement (A'): 1 - P(A) = 3/4

Odds

• Odds in Favor:
  • Calculated as: P(A) / P(A')
  • Example: If P(A) = 1/4 and P(A') = 3/4
    • Odds in favor = (1/4) / (3/4) = 1/3
    • Simplified to: 1 to 3
• Odds Against:
  • Calculated as: P(A') / P(A)
  • Example: Using the same P(A) and P(A') as above
    • Odds against = (3/4) / (1/4) = 3/1
    • Simplified to: 3 to 1

Example Problems

Example 1: Choosing a Face Card

• Scenario: Mario chooses a single card from a standard deck where all hearts are in Janara's hand
• Face Cards: Jack, Queen, King of hearts
• Total Hearts: 13
• Face Cards: 3
• Non-Face Cards: 10
• Odds in Favor:
  • Probability of face card = 3/13
  • Probability of non-face card = 10/13
  • Odds in favor = 3 to 10

Example 2: Day of the Week

• Scenario: What are the odds against a randomly chosen day being Sunday?
• Probability of Sunday: 1/7
• Probability of Not Sunday: 6/7
• Odds Against:
  • Odds against = 6/1
  • Which is: 6 to 1

Example 3: College Gift Card

• Scenario: Determining the probability from given odds
• Odds Against: 57 to 43 that a student is not in the first year
• Calculating Probability:
  • Adds up odds to get total outcomes: 57 + 43 = 100
  • Probability of being in the first year = 43/100 = 43%

Example 4: Hockey Shootout

• Scenario: Deciding who should take the shot based on past success rates
• Ellen: 8 goals out of 13 attempts (Probability: 0.615)
• Brittany: 10 goals out of 17 attempts (Probability: 0.588)
• Josie: 2 goals out of 3 attempts (Probability: 0.667)
• Decision: Based on highest probability or consistency (Ellen)

Example 5: Charity Carnival Games

• Scenario: Comparing odds against winning two games
• Bim: Odds against = 5 to 2 (converts to odds in favor = 2 to 5)
• Zap: Odds against = 7 to 3 (converts to odds in favor = 3 to 7)
• Probabilities:
  • Winning Bim: 2/7 = 0.286
  • Winning Zap: 3/10 = 0.3
• Decision: Higher probability of winning Zap

Summary

• Odds in Favor: P(A) / P(A')
• Odds Against: P(A') / P(A)
• Clarification: Odds are not the same as probability ratios; total outcomes cancel out in odds

Next Steps

• Upcoming Topics: Counting methods, set theory, and notation for solving more complicated probabilities

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