In an examination, the probability of a candidate solving a questions is `(1)/(2)` Out of given 5 question in the examination, what is the probability that the candidate was able to solve at least 2 questions?

Options

A. `(1)/(64)`
B. `(3)/(16)`
C. `(1)/(2)`
D. `(13)/(16)`

Explanation

To solve the problem of finding the probability that a candidate was able to solve at least 2 questions out of 5, we can follow these steps:

### Step 1: Define the problem
We need to find the probability that the candidate solves at least 2 questions out of 5. The probability of solving a question is given as \( p = \frac{1}{2} \).

### Step 2: Identify the complementary event
Instead of calculating the probability of solving at least 2 questions directly, we can find the probability of the complementary event (solving less than 2 questions) and subtract it from 1. The event of solving less than 2 questions includes solving 0 or 1 question.

### Step 3: Calculate the probability of solving 0 questions
Using the binomial probability formula:
\[ P(X = r) = \binom{n}{r} p^r (1-p)^{n-r} \]
where:
- \( n = 5 \) (total questions),
- \( r = 0 \) (number of questions solved),
- \( p = \frac{1}{2} \) (probability of solving a question),
- \( q = 1 - p = \frac{1}{2} \) (probability of not solving a question).

Calculating \( P(X = 0) \):
\[ P(X = 0) = \binom{5}{0} \left(\frac{1}{2}\right)^0 \left(\frac{1}{2}\right)^{5} = 1 \cdot 1 \cdot \frac{1}{32} = \frac{1}{32} \]

### Step 4: Calculate the probability of solving 1 question
Calculating \( P(X = 1) \):
\[ P(X = 1) = \binom{5}{1} \left(\frac{1}{2}\right)^1 \left(\frac{1}{2}\right)^{4} = 5 \cdot \frac{1}{2} \cdot \frac{1}{16} = \frac{5}{32} \]

### Step 5: Calculate the probability of solving less than 2 questions
Now, we can find the total probability of solving less than 2 questions:
\[ P(X < 2) = P(X = 0) + P(X = 1) = \frac{1}{32} + \frac{5}{32} = \frac{6}{32} = \frac{3}{16} \]

### Step 6: Calculate the probability of solving at least 2 questions
Using the complementary probability:
\[ P(X \geq 2) = 1 - P(X < 2) = 1 - \frac{3}{16} = \frac{16 - 3}{16} = \frac{13}{16} \]

### Final Answer
The probability that the candidate was able to solve at least 2 questions is:
\[ \frac{13}{16} \]

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