Question

If the random variable \(X\sim B(4,0.5)\), then probability \(P(X\lt 3/X\geq 1)=\ldots\)

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

Hint:

Use P(A|B) = P(A and B)/P(B) with the binomial probabilities for n = 4, p = 0.5.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept
For \(X\sim B(4,0.5)\), \(P(X=r)=\dbinom{4}{r}\left(\dfrac{1}{2}\right)^{4}=\dfrac{\dbinom{4}{r}}{16}\).

Step 2: Set up
\(P(X\lt3\mid X\geq1)=\dfrac{P(1\leq X\leq2)}{P(X\geq1)}\).

Step 3: Calculation
\(P(X=1)+P(X=2)=\dfrac{4+6}{16}=\dfrac{10}{16}\) and \(P(X\geq1)=1-\dfrac{1}{16}=\dfrac{15}{16}\).

Step 4: Result
The probability is \(\dfrac{10}{15}=\dfrac{2}{3}\), option (D).

Final Answer:
The conditional probability is 2/3. \[ \boxed{\text{(D) }\dfrac{2}{3}} \]