# probability problem

TDR1
Posers and Puzzles 13 Mar '05 23:26
1. 13 Mar '05 23:26
You and a friend are playing Scrabble, which involves lettered tiles. Here is the distribution of the tiles:

A:9 B:2 C:2 D:4 E:12 F:2 G:3
H:2 I:9 J:1 K:1 L:4 M:2 N:6
O:8 P:2 Q:1 R:6 S:4 T:6 U:4
V:2 W:2 X:1 Y:2 Z:1 Blank:2

At the start of the game, you choose seven letters. Solve the following, showing all work.

a) what is the probability that you will choose three vowels and four consonants? (counting 'Y' as a vowel)

b) what is the probability that you will choose the letters A,B,C,D,E,F, and G in order?

A calculator is going to be needed. Have Fun!
2. 14 Mar '05 05:034 edits
Originally posted by TDR1

a) what is the probability that you will choose three vowels and four consonants? (counting 'Y' as a vowel)

b) what is the probability that you will choose the letters A,B,C,D,E,F, and G in order.

There are
9 A's
12 E's
9 I's
8 O's
4 U's
2 Y's
So total vowels = 44

There are
100 tiles
2 Blanks
So total consonants = 98 - 44 = 54

a) Number of ways of arranging 3 vowels and 4 consonants = 7! / 3! 4! = 35
Number of ways of choosing 3 vowels and 4 consonants in a particular order = (44*43*42)*(54*53*52*51) / (100*98*97*96*95*94*93)
149586723 / 18796757000

Hence probability of choosing 3 vowels and 4 consonants in any order
149586723 / 537050200
= 0.2785339676

b) Required probability = 9*2*2*4*12*2*3 / (100*98*97*96*95*94*93)
= 1 / 723675144500
= 0.00000000000381835493

3. 14 Mar '05 09:282 edits
Originally posted by THUDandBLUNDER
Number of ways of choosing 3 vowels and 4 consonants in a particular order = (44*43*42)*(54*53*52*51) / (100*98*97*96*95*94*93)

Oops, that should be
Number of ways of choosing 3 vowels and 4 consonants in a particular order = (44*43*42)*(54*53*52*51) / (100*99*98*97*96*95*94)
= 4532931 / 606347000

Hence probability of choosing 3 vowels and 4 consonants in any order
= 4532931 / 17324200
= 0.2616531211

4. 15 Mar '05 02:25
But that doesn't account for the possibility of one or two of the seven tiles being blanks, which can be used as either vowels or consonants.
5. 15 Mar '05 03:19
Originally posted by Gyr
But that doesn't account for the possibility of one or two of the seven tiles being blanks, which can be used as either vowels or consonants.
I believe the question was about how the tiles are chosen, not about how they are played.