Presentation on theme: "Getting Married The Mathematical way The simulation game N numbers are written on N pieces of paper which are then shaken up in a box. The numbers are."— Presentation transcript:
The simulation game N numbers are written on N pieces of paper which are then shaken up in a box. The numbers are drawn one by one. We can stop when we like and take the last number … But we can’t go back 7 4 9
The objective We want to select the largest number (We want the strategy which gives us the best chance of choosing the largest number). i.e. We want to MAX the probability.
The problem We have no idea of the order in which the numbers will be drawn E.g. 7, 9, 4, 3, 5, 8, 6, 1, 3, 3, 0, 2 It is only when the pth number drawn is the biggest so far That we will even consider stopping.
Probabilities P(pth number drawn is highest so far) P(pth number is highest of all) P(pth is highest given highest so far)
Decision time Suppose there are t numbers left to be drawn. We have to decide whether …. (a) To stop on number just chosen, Or (b) To continue “sampling”.
Defining useful variables Let be the probability of winning if we continue (i.e. selecting the highest number of all). So, the probability of winning if we stop is
The objective variable Let be the probability of winning by adopting the best strategy when there are t numbers left. (1)
Considering the strategy Either the number we draw next is the highest so far AND We adopt the best strategy when (t -1) are left ………
Considering the strategy OR …. The number we draw next is not the highest so far AND We therefore must continue
The iteration formula For t > 1, We first need then (2)
Starting values From equation (1), (For N > 2) So when t = 1, STOP if i.e. Or N >2 and number just drawn is highest so far
Using the iteration formula From equation (2) when t = 2, So from (1),
Termination So when t = 2, STOP if And the number just drawn is the highest so far
More iterations In the same way we can obtain And of course Then, when t = 3 we STOP if And number drawn is highest so far
The optimum strategy When there are t numbers left, we should STOP if And the number just drawn is the highest so far
Example time Click here to see the exampleClick here to see the example Click here to skip the exampleClick here to skip the example
Example: Taking N = 8 1 left 2 left 3 left 4 left In each case above, we STOP if the number just drawn is highest so far
Example continued But when there are 5 left, And So we would continue Note that with 6 left, And with 7 left,
“Courting” R = N – t is the number we select for information and experience. N23456789 R11122233 N1011121314152025303540 R34455579111315 N455060708090100 R16182226293337
Outcome For large values of N, the probability of winning by adopting this strategy is Where t is the smallest integer such that