Luana is playing a game. She has 6 cards numbered 1 to 6. She has to place them into the three positions of this grid to make the highest possible total she can.
The right-most position represents the ones column for the number that she will finally produce. The middle position is the tens column and the left-most position is for the hundreds.
She has to position the cards after rolling a dice. She rolled a 3 first and decided to place card 3 into the tens column.
She then rolled a 5. Where should she place the 5 to get the highest 3-digit number on average? [She only has 1 of each card, if she rolls another 3, or another 5, she just rolls again.]
Solution
This can easily be solved by looking at all the possibilities for the third roll.
First - what happens if Luana puts the 5 in the ones column?
There are 4 possibilities for the next roll (635, 435, 235, 135). All are equally likely.
Next, what happens if she puts the 5 in the hundreds column?
The 4 possible results are 536, 534, 532, or 531.
If a 6 is thrown, 635 beats 536. However, if any other number is thrown, it is better to have the 5 in the hundreds column. ‘Any other number’ is thrown three times (1, 2, or 4). So on average, we would expect that the 5 in the hundreds’ column would win 3 times out of 4 (or with a probability of 3/4).
Extension
Can you come up with a set of rules for what card you should place where in any given situation?