In order to decide who will play ﬁrst in a certain board game, each player rolls a four-sided die; the player with the highest score begins.

(i) What is the probability that two particular people obtain the same score?

(ii) Three people decide to play the game. What is the probability that when they roll the die to decide who starts, they all obtain diﬀerent scores?

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i) If two people roll the dice you have 4^2 possible combinations. Of these combinations, 4 of them can be the same:

1 - 1

2 - 2

3 - 3

4 - 4

Therefore, the probability that the two people will roll the same number is:

`P_(same)=("#same")/("#all")=4/4^2=4/16=1/4`

ii) If three people roll the dice you have 4^3 possible combinations. Of these combinations, 4! are the number of possible rolls in which no player rolls the same. Therefore, the probability that no player rolls the same number is:

`P_("different")=("#different")/("#total")=(4!)/4^3=(4*3*2*1)/64=24/64=3/8`

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