Or Probabilities

• If two events cannot occur at the same time, they are mutually exclusive.
• Rolling a 2 and rolling a 3 on a six sided die in one roll.
• Rolling a 2 and rolling an odd number on a six sided die in one roll.
• Selecting a black card and selecting a heart from by selecting a single card from a deck.
• When two events (E₁ and E₂) are mutually exclusive
• P(E₁ or E₂) = P(E₁) + P(E₂)
• Examples

  Roll a single die, find the probability that the result is 2 or 3
     P(2) = 1/6
     P(3) = 1/6
     These events are mutually exclusive
     P(2 or 3) = 1/6 + 1/6 = 2/6 = 1/3
  
  Roll a single die, find the probability that the result is odd or a 2.
     P(2) = 1/6
     P(odd) = 3/6 
     These events are mutually exclusive
     P(2 or odd) = 1/6 + 3/6 = 4/6 = 2/3
  
  Select a single card from a deck.  What is the probability it is a heart or black?
     P(heart)  = 1/4
     P(black) = 2/4
     P(heart or black) = 1/4 + 2/4 = 3/4
      

• When two events (E₁ and E₂) are not mutually exclusive
• P(E₁ or E₂) = P(E₁) + P(E₂) - P(E₁ and E₂)
• Examples:

  Roll a single die, find the probability of even or 2
      P(2) = 1/6
      P(even) = 3/6
      P(2 and even) = 1/6
      P(2 or even) = 3/6 + 1/6 - 1/6 = 3/6 or 1/2
  
  Draw a single card, find the probability that it is red or a queen.
      P(red) = 26/52
      P(queen) = 4/52
      P(red and queen) = 2/52
      P(red or queen) = 26/52 + 4/52 - 2/52 = 28/52 = 7/13
      

Odds

• Odds are often used with games of chance.
• Odds, unless otherwise stated, are always odds against.
• The odds in favor of an event are expressed as
• P(E)/P(not E), or as P(E): P(not E)
• The odds against an event are
• P(not E)/P(E) or P(not E) : P(E)
• Examples:

  Find the odds in favor of rolling a 4 on a 6 sided die.
     P(E) = 1/6
     P(not E) = 5/6
     odds in favor = 1/6 / 5/6 = 1:5
     odds against = 5/6  / 1/6 = 5:1
  
  Find the odds against being dealt a queen
     P(E) = 4/52
     P(not E) = 48/52
     odds against are   48:4 or 12:1
     

• If we know the odds, we can compute the probability of an event.
• if the odds in favor of an event are a:b
• P(E) = a/(a+b)
• P(not E) = b/(a+b)
• Example:

  The odds against winning by selecting a number in roulette are 37 : 1  
  Find the probability of winning on a single number.
     P(E) = 1/(37+1) = 1/38
     

Homework