Chapter at a Glance

Every probability question is the same idea: count the equally likely outcomes, count the favourable ones, and divide. Use the complement whenever the wording says "not", "at least" or "none".

Key Facts and Formulas

  • Theoretical probability: P(E)=number of favourable outcomestotal number of equally likely outcomesP(E)=\dfrac{\text{number of favourable outcomes}}{\text{total number of equally likely outcomes}}.
  • Range: 0≤P(E)≤10\le P(E)\le 1.
  • Impossible event: P=0P=0. Sure event: P=1P=1.
  • Elementary events sum to 1.
  • Complement: P(E‾)=1−P(E)P(\overline{E})=1-P(E), i.e. P(E)+P(E‾)=1P(E)+P(\overline{E})=1.

Standard Sample Spaces (Memorise the Totals)

  • One coin: 2 outcomes {H,T}\{H,T\}. Two coins: 4 outcomes {HH,HT,TH,TT}\{HH,HT,TH,TT\}.
  • One die: 6 outcomes. Two dice: 36 outcomes; sum 7 is most likely (6 ways); 6 doublets.
  • Deck of cards: 52 = 4 suits ×\times 13. 26 red, 26 black; 13 of each suit; 12 face cards (J, Q, K); 4 aces; 2 red kings.
  • Bag of balls / spinner / numbers: P=favourable counttotal countP=\dfrac{\text{favourable count}}{\text{total count}}; equal sectors are equally likely.

Common Traps to Avoid

  • Probability stays in [0,1][0,1] — a negative answer or one above 1 means a counting error.
  • Two dice give 36 outcomes, not 12; and the sums are not equally likely.
  • "Greater than 4" excludes 4; "at least" includes the boundary — read the wording exactly.
  • Face cards are 12 (J, Q, K); do not forget or double-count the aces.
  • If items are removed first, update the total before dividing.
  • "Not / at least / none" — use the complement 1−P(E)1-P(E).

Final Tip: Write the total first, count favourable carefully, and sanity-check that your answer lies between 0 and 1.