Smart math sat people question plz

Discus: SAT/ACT Tests and Test Preparation: July 2004 Archive: Smart math sat people question plz
 By Hermit_Mmood (Hermit_Mmood) on Friday, July 30, 2004 - 02:50 pm: Edit

BEST EAST

A letter will be chosen at random from each of the two words above. What is the probability that the two letters chosen will be the same?
A) 3/32
B) 3/16
C) 1/4
D) 1/3
E) 3/8

there are two ways that I know to do it: 1) Just list them all out and get 3 repetitions out of 16 combinations or 2) 3/4 x 1/4 = 3/16

The 1st one is pretty self-explanatory but the 2nd one confuses me as to why you multiply 3 out of the 4 letters in the first word with only 1 out of the 4 letters in the second word.

this is more of the concept of probability, analyzing why you MULTIPLY the two numbers to get the probability. Why is this so?

 By Qwert271 (Qwert271) on Friday, July 30, 2004 - 02:53 pm: Edit

3 out of the 4 letters in BEST have 1 letter in common out of the 4 letters in EAST.

 By Hermit_Mmood (Hermit_Mmood) on Friday, July 30, 2004 - 02:57 pm: Edit

3 out of the 4 letters in BEST have 3/4 letters in EAST. This is wrong yes, but couldn't this be rationalized by someone who didn't think that?

 By Legendofmax (Legendofmax) on Friday, July 30, 2004 - 03:23 pm: Edit

When you have a case of (probability of this) AND (probability of that) it's the same as multiplying the two probabilities together. AND signifies multiplication, while OR signifies addition.

The probability we wish to find is this:
(probability of getting E and E) OR (probability of getting S and S) OR (probability of getting T and T).

BEST EAST

Now, the only letters that can be chosen twice from the two words are E, S, and T, since B is only in BEST and A is only in EAST.

So:

Probability of getting 1 letter in BEST: 1/4
Probability of getting 1 letter in EAST: 1/4

So, the probability of getting two E's is (1/4)(1/4) or (1/16). Probability of getting two S's is also the same, (1/4)(1/4) or (1/16), and likewise for two T's. Since there are three ways we can get two letters (this OR this OR this), we add the probabilities together.

1/16 + 1/16 + 1/16 = 3/16

 By Hermit_Mmood (Hermit_Mmood) on Saturday, July 31, 2004 - 12:43 am: Edit

thanks, that clarifies it a bit. So there really isnt foundation to multiplying or adding in probability? You just do it, without asking yourself why MULTIPLYING gets you the right answer it just works. Do you see what I mean? I guess i'm trying to prove to myself why you multiply to get the probability answer. What is the rationality behind multiplying.

 By Benzinspeicher (Benzinspeicher) on Saturday, July 31, 2004 - 01:00 am: Edit

for each of the four letters that is chosen from the first word, there are four letters that can be chosen from the second word, so 4 letters for every 4 letters, so 4 x 4 = 16 possibilities. If the two words were "Report Offensive", then the total possiblities would be 6 letters from the first word for each of the 9 letters from the second word, or 6 x 9 = 54 possiblities.

Now, there are three letters that both words have in common- "E", "S", and "T". Henceforth, the answer is 3/16.

 By Etsrep78328 (Etsrep78328) on Saturday, July 31, 2004 - 11:23 am: Edit

In terms of probability, in order to answer this question, you must consider the probability of getting an E and an E or S and S or T and T.

The word "and" signifies multiplication, and the word "or" signifies addition in probability/combinatorics (usually).

P(E and E) + P(S and S) + P (T and T) = (1/4 * 1/4) + (1/4 * 1/4) + (1/4 * 1/4) = 1/16 * 3 = 3/16.