What is wrong with this?

Probability Level 3

Alexander wants to find the probability that when 10 people were to each roll a regular 6-sided die, the first player has the highest value. He thinks that the answer is $\frac1{512}$ , and has come up with the following solution:

Step 1 . This is equivalent to the first player out-rolling the second player, the first player out-rolling the third player, and so on.

Step 2 . The probability that the first player rolls a higher number than the second does is $\frac{1}{2}$ , and same for the other players.

Step 3 . Thus, the overall probability is $\left(\frac{1}{2}\right)^9={\frac{1}{512}}$ .

Which of these 3 steps is wrong?

Note: When evaluating a step, you should assume that the other steps are valid.

Step 2 Nothing is wrong Steps 1 and 2 Steps 1 Step 3 Steps 1, 2 and 3 Steps 1 and 3 Steps 2 and 3

1 solution

Alex Li
Jul 29, 2015

Steps 2 and 3 are both incorrect.

Step 2 is incorrect because the probability of the first player out-rolling any given other player is not $\frac{1}{2}$ , because there is a possibility of equal rolls.

Step 3 is also incorrect because the events of out-rolling the players individually are not independent. Given that Player 1 out-rolled Player 2, it is more likely that Player 1 rolled a higher number, so it is more likely that he out-rolled Player 3 than if not given that information. Thus, we cannot simply raise the probability of winning an individual roll to the 9th power.

Moderator note:

Great analysis! Those are 2 very common mistakes that are made.

What would the actual probability be?

Great question! I made several edits to it, can you check that the problem is still correct?
- Edited the problem
- Edited the choices.

Calvin Lin Staff - 5 years, 10 months ago

The problem is still correct as intended. The actual probability would be $\frac{(\frac{1}{6})^9+(\frac{2}{6})^9+...+(\frac{5}{6})^9}{6}$

Alex Li - 5 years, 10 months ago

Close, but not quite. Remember that the dice only have 6 sides.

Calvin Lin Staff - 5 years, 10 months ago

I agree with solution with the assumption that the first gets the highest roll and none of the others equals the roll. Now, getting the highest roll must be defined. Is it enough to be said the highest when none is higher? Does it have to be higher than a certain number of rolls? Does it have to be higher than all 9 rolls.

Roman Frago - 5 years, 10 months ago

What is wrong with this?

1 solution

Moderator note:

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