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Paul and Jesse each choose a number at random from the first six primes. What is the probability that the sum of the numbers they choose is divisible by 3?

 Apr 16, 2022

Best Answer 

 #2
avatar+9369 
+1

Here are the first 6 primes: \(\{2,3,5,7,11,13\}\)

 

We can take a look at all the possibilities by making a table. 

 

\(\begin{matrix}&\color{blue}2&\color{blue}3&\color{blue}5&\color{blue}7&\color{blue}11&\color{blue}13\\\color{red}2&4&5&7&\color{magenta}9&13&\color{magenta}15\\\color{red}3&5&\color{magenta}6&8&10&14&16\\\color{red}5&7&8&10&\color{magenta}12&16&\color{magenta}18\\\color{red}7&\color{magenta}9&10&\color{magenta}12&14&\color{magenta}18&20\\\color{red}11&13&14&16&\color{magenta}18&22&\color{magenta}24\\\color{red}13&\color{magenta}15&16&\color{magenta}18&20&\color{magenta}24&26\end{matrix}\)

 

Blue numbers and red numbers represent the number that Paul chose and the number that Jesse chose respectively.

Pink numbers are those divisible by 3.

 

By trial-and-error, out of all 36 possible cases, 13 cases has a sum divisible by 3.

 

Therefore, 

 

\(\text{Prob}(\text{sum divisible by 3}) = \dfrac{13}{36}\)

 Apr 16, 2022
 #1
avatar+36417 
+1

Assuming they can pick the same number

 

36 choices

2 7

2 13

5 7

5 13

7 11

3 3              the red ones can be reversed      11 choices out of 36

 

OOps...I see from Max answer I missed   11 13      13 11     so the answer would be the same    13 /36          (Thanx, max!)

 Apr 16, 2022
edited by Guest  Apr 16, 2022
 #2
avatar+9369 
+1
Best Answer

Here are the first 6 primes: \(\{2,3,5,7,11,13\}\)

 

We can take a look at all the possibilities by making a table. 

 

\(\begin{matrix}&\color{blue}2&\color{blue}3&\color{blue}5&\color{blue}7&\color{blue}11&\color{blue}13\\\color{red}2&4&5&7&\color{magenta}9&13&\color{magenta}15\\\color{red}3&5&\color{magenta}6&8&10&14&16\\\color{red}5&7&8&10&\color{magenta}12&16&\color{magenta}18\\\color{red}7&\color{magenta}9&10&\color{magenta}12&14&\color{magenta}18&20\\\color{red}11&13&14&16&\color{magenta}18&22&\color{magenta}24\\\color{red}13&\color{magenta}15&16&\color{magenta}18&20&\color{magenta}24&26\end{matrix}\)

 

Blue numbers and red numbers represent the number that Paul chose and the number that Jesse chose respectively.

Pink numbers are those divisible by 3.

 

By trial-and-error, out of all 36 possible cases, 13 cases has a sum divisible by 3.

 

Therefore, 

 

\(\text{Prob}(\text{sum divisible by 3}) = \dfrac{13}{36}\)

MaxWong Apr 16, 2022

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