Events Involving And

• Two events are called independent if the occurrence of either has no impact on the other.
• Toss a coin twice
• Roll two different dice
• flip a coin and draw a card.
• For independent events, P(E₁ and E₂)= P(E₁)·P(E₂)
• Two die are tossed, find the probability of rolling two 2s.

  P(2) = 1/6
  P(2) = 1/6
  P(2 and 2) = 1/36
      

• A coin is flipped and a die is tossed. Find the probability of a head and the number is less than three.

  P(heads) = 1/2
  P(less than three) = 2/6
  P(heads and less than three) = 1/2 * 2/6 = 1/6
      

• A card is drawn and a die is tossed. Find the probability of a red card and an odd number.

  P(red) = 1/2
  P(odd) = 1/2
  P(red and odd) = 1/2  * 1/2 = 1/4
      

• Assume the sex of a child is independent. Find the probability of two girls in a row, three?

  P(E) = 1/2
  P(E and E) = 1/2 * 1/2 = 1/4
  P(E and E and E) = 1/4 * 1/2 = 1/8
      

• If a family has three girls in a row, what is the probability that the next child is a girl?

  P(E) = 1/2
      

• Two events are dependent if the occurrence of one has an impact on the occurrence of the other.
• If you draw two cards from a deck, what is the probability of getting two kings.
• P(E₁ and E₂) for dependent events is
• P(E₁)·P(E₂ given that E₁ Occurred)
• Two kings

  P(1 king) = 4/52
  P(1 king given 1 king) = 3/51
  P(one king and one king) = 4/52 * 3/51
  
      

• This is sometimes called sampling without replacement.

Conditional Probability

• What is the probability of rolling a two if we know a even number was rolled?
• P(A|B) probability of event A occurring given that event B occurred.
• P(A|B_ = n(A and B) / N(B)
• P(2 given even) = 1/3

Homework