0580

Basic Probability

Probability

Sample space diagrams

  • Where two things happen together, such as two dice or a dice and a coin, a table lays out every combination
  • Put one experiment's outcomes along the top and the other's down the side, then fill in the result of each pair
  • The number of cells is the total number of outcomes, and counting the ones that satisfy the event gives the numerator
  • Two ordinary dice give 6 × 6 = 36 outcomes, and a systematic table makes them impossible to miscount
    • A total of 9 arises from (3, 6), (4, 5), (5, 4) and (6, 3), so its probability is 4 ÷ 36 = 1/9
A 6 by 6 table with the first dice down the side and the second across the top, each of the 36 cells holding the total of that pair, so the totals run from 2 in the top left corner to 12 in the bottom right
Source: Possibility Diagrams by Save My Exams
  • Order usually matters, so (3, 6) and (6, 3) count separately unless the question says otherwise
  • Filling the whole table before counting is faster than trying to spot the successes directly
The same table of dice totals with every odd total above 6 circled: six 7s, four 9s and two 11s, twelve circled cells out of the 36, which is how the numerator of the probability is counted
Source: Possibility Diagrams by Save My Exams
Worked example

Reading a probability off a sample space

Two ordinary dice are rolled and their scores added. Find the probability that the total is 10 or more.

Solution:

  • Each die has 6 faces, so the sample space has 6 × 6 = 36 equally likely outcomes
  • List the pairs giving a total of 10 or more, taking them in order so none is missed
  • Total 10: (4, 6), (5, 5), (6, 4)
  • Total 11: (5, 6), (6, 5)
  • Total 12: (6, 6)
  • That is 6 successful outcomes out of 36
  • P(total ⩾ 10) = 6/36 = 1/6
  • Order matters here, so (4, 6) and (6, 4) are two separate outcomes, not one