MATH 109
The Name Game
In the Name Game, the letters of the alphabet are drawn at random and players mark off
each occurrence of the letter in their name as it is called. The winner is the player
whose name is deleted first; but a tie could occur when the players have letters in
common. For a two-player game, the probability of a player winning depends on how
many letters occur only in the other player’s name.
For example, if Antonio plays against Gabriella, then there are 10 letters in the
union, 2 letters in the intersection, 3 letters that are in Antonio but not in Gabriella, and
5 letters that are in Gabriella but not in Antonio. Only the 10 letters in the union affect
the game, and a single game really consists of a permutation of these 10 letters since all
other letters can be ignored.
A tie occurs when the last letter chosen comes from the intersection and exhausts
both names at the same time. So there are 2 choices out of 10 for the last letter to cause a
tie. Antonio wins if the last letter is in Gabriella but not in Antonio which gives 5
choices; and Gabriella wins if the last letter is in Antonio but not in Gabriella which
gives 3 choices.
Antonio (Set A) vs. Gabriella (Set G)
(Antonio): A = {a, n, t, o, i}
(Gabriella): G = {g, a, b, r, i, e, l}
A
G
n
g
a
b
t
i
o
r
e
l
c d f h j k m p q s u v w x
y z
n(A) = 5
n(G) = 7
n(A − G) = 3
A
A′
n(A ∪ G) = 10
n(G − A) = 5
G
2
5
7
n(A ∩ G) = 2
n( A ′ ∩ G′) = 16
G′
3
16
19
5
21
26
There are 10 letters in the union, 2 in the intersection. So the probability of a tie is
2 / 10 . The probability of Antonio winning is 5 / 10 (the 5 comes from the extra letters in
the other person’s name). And the probability of Gabriella winning is 3 / 10 .
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Activity
(i) Choose a partner in class.
(ii) Label each other with a distinct set symbol such as A or B .
(iii) Label a blank sheet of paper as “My Name ( A ) vs. Partner’s Name ( B )” using you
and your partner’s names and symbols.
(iv) List the elements in the two sets, labeled as your symbols, that contain the distinct
letters of each of your names.
(v) Make a Venn Diagram showing the letters of your two sets along with the entire
alphabet.
(vi) Make a Block diagram that shows the number of elements in your two sets.
(vii) When playing the Name Game with your partner, give the probability of there
being a tie, the probability that you win, and the probability that your partner wins.
(viii) The Name Game will now be played in class with the Professor calling out letters
at random. Who won, you or your partner? Was this win an upset? How many letters
needed to be called for the game to end?
(ix) The average number of letters needed to be called to complete a game depends on
the number of letters N in the entire alphabet being used, and the number of letters in
the sets A and B of the players names. This average is given by
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 A
B
A∪B 
(N + 1) × 
+
−
.
 A + 1 B + 1 A ∪ B + 1
For you and your partner’s names, what is the average number of letters needed to be
called to complete a game? How does the actual number of letters needed in Part (viii)
compare to this theoretical average?