MHB Multi part Question involving probability and combinatrics with words

  • Thread starter Thread starter hb2325
  • Start date Start date
  • Tags Tags
    Probability
AI Thread Summary
The discussion revolves around calculating probabilities related to the arrangement of the letters in the word "TARANTULA." Participants explore various scenarios, including the likelihood of the first three letters containing no A's, the fourth letter being the first A, and the first and last letters being the same. There is a consensus that the probabilities for the first three letters and the last three letters are identical, while clarifications are sought regarding the correct approach to calculating these probabilities. Some participants express uncertainty about their calculations, particularly regarding the total permutations of "TARANTULA," indicating a need for a better understanding of combinatorial principles.
hb2325
Messages
19
Reaction score
0
4. An experiment consists of randomly rearranging the 9 letters of the word
TARANTULA into a sequence of 9 letters, where all possible orders of these 9 letters are equally
likely. Find the probability of each of the following events:

( a ) the first three letters include no A's;
( b ) the first three letters or the last three letters (or both) include no A's;
( c ) the fourth letter is the first A;
( d ) the first letter and the last letter are the same;
( e ) the word `TARANTULA' is obtained;
( f ) the sequence contains the word `RAT'.Attempt at solutions :

a) 6/9 * 5/8 * 4/7 (Probability of first non A * another non A letter * another non A letter)

b) 6/9 * 5/8 * 4/7 + ( I am stuck I don't get it - I think it might just be 2 * anser of part a but I'm unable to think it through, I know though that if first 3 and last 3 have no A's, then middle will have all A's so it becomes more weird )

c) 6/9 * 5/8 * 4/7 * 3/6 ( Prob. in a * probability of 4th letter being A in the scenario of question a)

d) 5/9 * 1/8 + 5/9 * 2/8 (Probability of choosing a repeatable letter * probability of second letter coming up for both T and A since R U L N are not repeatable)

e) 1/9!

f) After working it out on paper it seems there are 6 permutations of RAT so 6 * 1/9!
I am still not sure if these are correct and I know there must be a better way of doing these using combination formula so I would be greatful if someone could help me out.

Thanks!
 
Last edited by a moderator:
Physics news on Phys.org
hb2325 said:
4. An experiment consists of randomly rearranging the 9 letters of the word
TARANTULA into a sequence of 9 letters, where all possible orders of these 9 letters are equally
likely. Find the probability of each of the following events:

( a ) the first three letters include no A's;
( b ) the first three letters or the last three letters (or both) include no A's;
( c ) the fourth letter is the first A;
( d ) the first letter and the last letter are the same;
( e ) the word `TARANTULA' is obtained;
( f ) the sequence contains the word `RAT'.Attempt at solutions :

a) 6/9 * 5/8 * 4/7 (Probability of first non A * another non A letter * another non A letter)

b) 6/9 * 5/8 * 4/7 + ( I am stuck I don't get it - I think it might just be 2 * anser of part a but I'm unable to think it through, I know though that if first 3 and last 3 have no A's, then middle will have all A's so it becomes more weird )

c) 6/9 * 5/8 * 4/7 * 3/6 ( Prob. in a * probability of 4th letter being A in the scenario of question a)

d) 5/9 * 1/8 + 5/9 * 2/8 (Probability of choosing a repeatable letter * probability of second letter coming up for both T and A since R U L N are not repeatable)

e) 1/9!

f) After working it out on paper it seems there are 6 permutations of RAT so 6 * 1/9!
I am still not sure if these are correct and I know there must be a better way of doing these using combination formula so I would be greatful if someone could help me out.

Thanks!


For ( b ) the required probability is:

P(first 3 contain no A's) + P(last 3 contain no A's) - P(both the first and last 3 contain no A's)

A Venn diagram will show that the sum of the first two terms counts "both the first and last 3 contain no A's" twice, so that probability needs to be removed once.

Your answers to ( a) and ( c) look OK.

( e) is wrong since there are 6x2 (since you have 3 "A"s and 2 "T"s in "TARANTULA") of the 9! permutations that give "TARANTULA".( d) probability that the first letter and last letter are A is (3/9)x(2/8), and that they are T is (2/9)x(1/8)

CB
 
Last edited:
Firstly thanks for taking the time to help me.

About b) The problem is I don't understand what P(Last 3 contain no A's) would be in terms of say how first 3 is 6/9 * whatever and so on? Would it also be 6/9*5/8*4/7? I'm kinda stumped on how to take it.

Also Part e) Yeah I overlooked the 3 A's and 2 T's part so still wouldn't it be 3x2 and not 6x2 :S Where does the 6 come from?

Thank
 
hb2325 said:
Firstly thanks for taking the time to help me.

About b) The problem is I don't understand what P(Last 3 contain no A's) would be in terms of say how first 3 is 6/9 * whatever and so on? Would it also be 6/9*5/8*4/7? I'm kinda stumped on how to take it.

The probability that the last 3 contain no A's is the same as the probability that the first three contain no A's

CB
 
I was reading documentation about the soundness and completeness of logic formal systems. Consider the following $$\vdash_S \phi$$ where ##S## is the proof-system making part the formal system and ##\phi## is a wff (well formed formula) of the formal language. Note the blank on left of the turnstile symbol ##\vdash_S##, as far as I can tell it actually represents the empty set. So what does it mean ? I guess it actually means ##\phi## is a theorem of the formal system, i.e. there is a...
Back
Top