Finding the highest number
With the strategy described, what is the probability that you will select the
biggest number?
The probability that the biggest number was on one of the first ten cards is
10
.
30
If this is the case then the probability you will select the biggest number is 0.
Now imagine that the biggest number happens to be on the 11th card; this has
1
probability
. Then you will select it (because it is bigger than the previous
30
1
10) and the probability that this is the scenario is
30
What if the biggest number is on card number 12? Then you require that the
biggest number on the first 11 cards was on one of the first 10 and not on the
11th card.
10
. Therefore the probability that the
The probability that this is the case is
11
1 10
biggest number is on card number 12 and also that you select it is
 .
30 11
Similarly, the probability that the biggest number is on card number 13 and
1 10
that you select it is
 .
30 12
Continuing in this way you can see that the probability that you select the card
containing the biggest number is
 10  1  1 10   1 10   1 10 
 1 10 
  0                ...     .
 30  30  30 11   30 12   30 13 
 30 29 
This can be simplified to
10  1 1 1 1
1 
     ...   .
30  10 11 12 13
29 
Using a spreadsheet, this value is found to be approximately 0.377.
You will find that, if you had decided in advance to turn over 11 (rather than
10) cards and to start making decisions from the 12th card then the probability
increases.
Investigate the optimal number of cards to turn over before starting to make
decisions for N cards.
© MEI 2008