Flipping Coins

Back to Probability Theory


  • A coin is randomly flipped three times in a row; what’s the probability of three heads?
  • Three coins are tossed simultaneously; what’s the probability of three heads?


Three Heads in a Row
  • The branches of the Possibility Tree represent the eight possible outcomes of flipping a coin three times, the rightmost branch representing three heads.  Thus the probability of tossing three heads is 1/8, one branch of eight.
Possibility Tree
  • The possibilities can also be listed in a Possibility Table:
Possibility Table
  • Rows in the table correspond to branches in the tree.
  • Finally, the probability can be computed using the Special Conjunction Rule (SCR):
    • P(A&B) = P(A) x P(B), where A and B are independent, i.e. where the truth of one doesn’t affect the probability of the other.
  • Thus the probability of three heads is:
    • P(H1 & H2 & H3) = P(H1) x P(H2) x P(H3) = ½ x ½ x ½ = ⅛
Three Heads Simultaneously
  • Tossing three coins simultaneously is logically analogous to tossing one coin three times, the situations differing in no way affecting their probabilities. The probability is thus ⅛.  
  • Argument by Analogy is useful in calculating probabilities in complex situations: think up a simpler, logically analogous scenario and calculate the corresponding probability.