136
Systems for Planning and Control in Manufacturing
10.2.5 Probability trees So far, only a maximum of two events has been discussed. In more complex situations it is useful to use probability trees. Consider four coins being tossed in succession. As shown in Figure 10.2 in this case there are 16 possible outcomes: Note that at each branching point, the probabilities add up to one in accordance with equation (10.2).
Figure 10.2 Probability tree for four coins
Key: H = Head
T = Tail
10.3 Probability distributions 10.3.1 General theory In Figure 10.2, if a head is worth 1 and a tail 0, then there is only one way to score 4 (HHHH) hence a probability of 1/16. Similarly, there is only one way to score 0 (TTTT). There are six ways, however, to score 2 (HHTT, HTHT, THTH, THHT, HTTH and TTHH) hence a probability of 6/16. If the score is plotted against probability, a probability distribution is produced as shown in Figure 10.3. The concept of a probability distribution is extremely valuable. It allows the probability of a particular outcome to be determined without the need to generate the probability tree.
10.3.2 The normal distribution There are a number of standard probability distributions. The one depicted in Figure 10.3 is a binomial distribution. The commonest type