Question:
If X is a binomial variable with range \(\{0,1,2,3,4\}\) and \(\mathrm{P}(\mathrm{X}=3)=3 \mathrm{P}(\mathrm{X}=4)\) then the parameter ' \(p\) ' of the binomial distribution is
If X is a binomial variable with range \(\{0,1,2,3,4\}\) and \(\mathrm{P}(\mathrm{X}=3)=3 \mathrm{P}(\mathrm{X}=4)\) then the parameter ' \(p\) ' of the binomial distribution is
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The range 0 to 4 shows n = 4; then equate the binomial probabilities.
Updated On: Nov 13, 2026
- \(\frac{1}{4}\)
- \(\frac{3}{4}\)
- \(\frac{1}{3}\)
- \(\frac{4}{7}\)
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The Correct Option is D
Solution and Explanation
Step 1: Understanding the Concept
A binomial variable with range \(\{0,1,2,3,4\}\) has \(n=4\), and \(P(X=r)=\dbinom{4}{r}p^{r}q^{4-r}\) with \(q=1-p\).
Step 2: Set up
\(P(3)=4p^{3}q\) and \(P(4)=p^{4}\), so \(4p^{3}q=3p^{4}\).
Step 3: Simplify
Dividing by \(p^{3}\): \(4q=3p\), so \(4(1-p)=3p\).
Step 4: Calculation
\(4=7p\), so \(p=\dfrac{4}{7}\).
Final Answer:
The parameter p is \(\frac{4}{7}\). \[ \boxed{\text{(D) }\dfrac{4}{7}} \]
A binomial variable with range \(\{0,1,2,3,4\}\) has \(n=4\), and \(P(X=r)=\dbinom{4}{r}p^{r}q^{4-r}\) with \(q=1-p\).
Step 2: Set up
\(P(3)=4p^{3}q\) and \(P(4)=p^{4}\), so \(4p^{3}q=3p^{4}\).
Step 3: Simplify
Dividing by \(p^{3}\): \(4q=3p\), so \(4(1-p)=3p\).
Step 4: Calculation
\(4=7p\), so \(p=\dfrac{4}{7}\).
Final Answer:
The parameter p is \(\frac{4}{7}\). \[ \boxed{\text{(D) }\dfrac{4}{7}} \]
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