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A machine randomly generates one of the nine numbers      with equal likelihood. What is the probability that when Tsuni uses this machine to generate four numbers their product is divisible by  Express your answer as a common fraction.

 Aug 15, 2024
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You are missing some values in your question, but I'm gonna assume the following was asked. 

 

A machine randomly generates one of the nine numbers   1,2,3...8,9   with equal likelihood. What is the probability that when Tsuni uses this machine to generate four numbers their product is divisible by 4. Express your answer as a common fraction.

 

________________________________________________________________________________________________________________________

 

Now, since we can draw two of the same numbers and order doesn't matter, we don't do (94) to find the total amount of possibilites, but instead just do

9999=94=6561

 

Now, we can use complementary counting to find the number of cases that DO NOT work and then subtract it from 1. 

This is because it's MUCH easier to calculate odd numbers rather than even numbers. 

 

Our first case is that all 4 numbers are odd. If all 4 numbers are odd, the product is odd, therefore meaning it cannot possibly be divisble by 4. 

Since out of the 9 numbers, 5 numbers are odd, we have a 5/9 chance each generation that the number is odd. 

Since we need 4 odd numbers in a row, we simply have (59)4=6256561

 

Our next case is where we get 3 odd numbers in a row and then a 2 or a 6. If we roll a 4 or 8, the product will automatically be divisble by 4. 

Thus, we have 2×53×(43)6561=10006561

 

Subtracting the sum from 1, we get the final answer of 1(6256561+10006561)=49366561

 

I would like verification of the final answer, but I believe this is the correct answer. 

 

Thanks! :)

 Aug 16, 2024
edited by NotThatSmart  Aug 16, 2024

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