Independence Problem: 3 Digits & Probability of Sending 0

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SUMMARY

The discussion focuses on the independence of two events related to the transmission of three binary digits (0 and 1) from a server to a computer. The events in question are A, defined as "at least 2 of 3 digits are 0," and B, defined as "all digits are the same." The conclusion reached is that these events are independent when the probability p of sending a 0 is either 1 or 1/2. The calculations involved determining the formulas for P(AB) and P(A)P(B) as functions of p and equating them to find the values of p.

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bour1992
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A server sends 3 digits (0 and 1) to a computer. The probability of sending 0 is p.
I have to check under which conditions the events:

A={at least 2 of 3 digits is 0}
B={all the digits are the same}

are independent.

I thought that they will be independent if the first 2 digits are different with each other.
e.g (0,1,0) or (1,0,0)

But i am not sure for that answer.
Can you tell me your ideas?
 
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If they are independent then P(AB) = P(A) P(B).
So to solve:
1. Determine a formula for P(AB) as a function of p
2. Determine a formula for P(A) P(B) as a function of p
3. Set them equal, and solve for p
 
thank you mXSCNT for your help.

So after calculations I found that p=1 or p=1/2.
Can you confirm that this is the right answer?
 

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