Question:
If \(P(X)\) and \(F(X)\) are the probability mass function and cumulative distribution function of discrete random variable \(X\) where \(X=\{0,1,2,3,\ldots ,10\}\) and \(a,b\) are integers such that \(0\lt a\lt b\lt 10\), then the conditional probability \(P(X\geq a/X\leq b)\) is equal to ...
If \(P(X)\) and \(F(X)\) are the probability mass function and cumulative distribution function of discrete random variable \(X\) where \(X=\{0,1,2,3,\ldots ,10\}\) and \(a,b\) are integers such that \(0\lt a\lt b\lt 10\), then the conditional probability \(P(X\geq a/X\leq b)\) is equal to ...
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Write the conditional probability as P(a <= X <= b) divided by P(X <= b) and use F.
Updated On: Nov 12, 2026
- \(1-\frac{F(a-1)}{F(b)}\)
- \(1-\frac{F(a)}{F(b)}\)
- \(1-\frac{F(a)}{F(b-1)}\)
- \(1-\frac{F(a-1)}{F(b-1)}\)
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The Correct Option is A
Solution and Explanation
Step 1: Understanding the Concept
\(F(k)=P(X\leq k)\), and for integers \(P(a\leq X\leq b)=F(b)-F(a-1)\).
Step 2: Apply the formula
\(P(X\geq a\mid X\leq b)=\dfrac{P(a\leq X\leq b)}{P(X\leq b)}\).
Step 3: Substitute
\(=\dfrac{F(b)-F(a-1)}{F(b)}=1-\dfrac{F(a-1)}{F(b)}\).
Step 4: Compare the options
This is the first option.
Final Answer:
The probability is \(1-\dfrac{F(a-1)}{F(b)}\). \[ \boxed{\text{(A) }1-\dfrac{F(a-1)}{F(b)}} \]
\(F(k)=P(X\leq k)\), and for integers \(P(a\leq X\leq b)=F(b)-F(a-1)\).
Step 2: Apply the formula
\(P(X\geq a\mid X\leq b)=\dfrac{P(a\leq X\leq b)}{P(X\leq b)}\).
Step 3: Substitute
\(=\dfrac{F(b)-F(a-1)}{F(b)}=1-\dfrac{F(a-1)}{F(b)}\).
Step 4: Compare the options
This is the first option.
Final Answer:
The probability is \(1-\dfrac{F(a-1)}{F(b)}\). \[ \boxed{\text{(A) }1-\dfrac{F(a-1)}{F(b)}} \]
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