Most likely tiles to draw from the bag?

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ScrabbleKitty

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Jan 2, 2020
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Trying to figure out how to calculate what combination of 2 tiles would be the most likely to be drawn in a game of Scrabble from a bag containing:

9 As
12 Es


This is what I am thinking,

12 in 21 chance of drawing an E first * an 11 in 20 chance of drawing an E second = 121/420 of drawing EE.
12 in 21 chance of drawing an E first * 9 in 20 chance of drawing an A second = 108/420 of drawing E then A
9 in 21 chance of drawing an A first * 12 in 20 chance of drawing an E second = 108/420 of drawing A then E
9 in 21 chance of drawing an A first * 8 in 20 chance of drawing an A second = 72/420 of drawing AA

so, drawing AE (not necessarily in that order) is most likely at 18/35 or approx. 51%
drawing EE coming in second at 121/420 or approx 29%
and finally AA at 6/35 or approx. 17%

I feel like maybe I'm doing something wrong because these add up to 409/420 instead of 420/420.


What am I doing wrong or what am I missing?
 
Hello, and welcome to FMH! :)

EE: [MATH]\frac{12}{21}\cdot\frac{11}{20}=\frac{11}{35}[/MATH]
EA: [MATH]\frac{12}{21}\cdot\frac{9}{20}=\frac{9}{35}[/MATH]
AE: [MATH]\frac{9}{21}\cdot\frac{12}{20}=\frac{9}{35}[/MATH]
AA: [MATH]\frac{9}{21}\cdot\frac{8}{20}=\frac{6}{35}[/MATH]
These add to 1.
 
ScrabbleKitty said:
12 in 21 chance of drawing an E first * an 11 in 20 chance of drawing an E second = 132/420 of drawing EE.
12 in 21 chance of drawing an E first * 9 in 20 chance of drawing an A second = 108/420 of drawing E then A
9 in 21 chance of drawing an A first * 12 in 20 chance of drawing an E second = 108/420 of drawing A then E
9 in 21 chance of drawing an A first * 8 in 20 chance of drawing an A second = 72/420 of drawing AA

so, drawing AE (not necessarily in that order) is most likely at 18/35 or approx. 51%
drawing EE coming in second at 132/420 or approx 31%
and finally AA at 6/35 or approx. 17%

I feel like maybe I'm doing something wrong because these add up to 409/420 instead of 420/420.

What am I doing wrong or what am I missing?
Click to expand...

See corrections above -- just one wrong multiplication.
 
yikes. I thought I new my 12 times tables. 11*11 is 121 is what I was thinking, don't know why I thought 12*11 was 121. Thanks a bunch. So I had the right strategy otherwise, then?
 
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