Question

If \(X\) is a binomial variable with the range \(\{0,1,2,3,4,5,6\}\) and \(P(X=2)=4P(X=4)\), then the parameter \(p\) of \(X\) is

• A. \(\frac{1}{3}\)
• B. \(\frac{1}{2}\)
• C. \(\frac{2}{3}\)
• D. \(\frac{3}{4}\)

Hint:

For binomial probabilities, use ratio method: cancel common factors and reduce to a simple equation in \(p\).

Answer 1

The Correct Option is A

Step 1: Identify \(n\).
Since range is \(\{0,1,2,3,4,5,6\}\), 
\[ n=6 \] 
Step 2: Write probability expressions. 
\[ P(X=2)=\binom{6}{2}p^2(1-p)^4 \] 
\[ P(X=4)=\binom{6}{4}p^4(1-p)^2 \] 
Step 3: Use given relation. 
\[ \binom{6}{2}p^2(1-p)^4 = 4\binom{6}{4}p^4(1-p)^2 \] 
Step 4: Simplify. 
\[ 15p^2(1-p)^4 = 4\cdot 15p^4(1-p)^2 \] 
Cancel \(15p^2(1-p)^2\): 
\[ (1-p)^2 = 4p^2 \] 
Take positive root (since \(p>0\)): 
\[ 1-p = 2p \Rightarrow 1=3p \Rightarrow p=\frac{1}{3} \] 
Final Answer: 
\[ \boxed{\frac{1}{3}} \] 
 

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