An exam consists of 3 problems selected randomly from a collection of 10 problems. For a student to pass, he needs to solve correctly at least two of three problems. If the student knows to solve exactly 5 problems, then the probability that the students pass the exam is

Options

A. `(1)/(2)`
B. `(1)/(3)`
C. `(3)/(4)`
D. `(5)/(6)`

Answer

To find the probability that a student passes the exam by solving at least 2 out of 3 problems correctly, we can break down the problem into two cases:

1. The student solves exactly 2 problems correctly and 1 problem incorrectly.
2. The student solves all 3 problems correctly.

### Step 1: Define the total number of problems and the problems the student can solve.
- Total problems = 10
- Problems the student can solve = 5
- Problems the student cannot solve = 5

### Step 2: Calculate the total ways to choose 3 problems from 10.
The total number of ways to select 3 problems from 10 is given by the combination formula \( \binom{n}{r} \), which is calculated as:
\[ \binom{10}{3} = \frac{10!}{3!(10-3)!} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120 \]

### Step 3: Calculate the probability of solving exactly 2 problems correctly and 1 incorrectly.
- To solve 2 problems correctly, the student can choose 2 from the 5 problems he knows how to solve:
\[ \binom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{5 \times 4}{2 \times 1} = 10 \]
- To solve 1 problem incorrectly, the student can choose 1 from the 5 problems he does not know how to solve:
\[ \binom{5}{1} = 5 \]
- Therefore, the total ways to select 2 correct and 1 incorrect problem is:
\[ \binom{5}{2} \times \binom{5}{1} = 10 \times 5 = 50 \]

### Step 4: Calculate the probability of solving all 3 problems correctly.
- To solve 3 problems correctly, the student can choose all 3 from the 5 problems he knows how to solve:
\[ \binom{5}{3} = \frac{5!}{3!(5-3)!} = \frac{5 \times 4}{2 \times 1} = 10 \]

### Step 5: Calculate the total number of successful outcomes.
The total number of successful outcomes (passing the exam) is the sum of the two cases:
\[ \text{Total successful outcomes} = \text{Case 1} + \text{Case 2} = 50 + 10 = 60 \]

### Step 6: Calculate the probability of passing the exam.
The probability of passing the exam is given by the ratio of successful outcomes to total outcomes:
\[ P(\text{pass}) = \frac{\text{Total successful outcomes}}{\text{Total ways to choose 3 problems}} = \frac{60}{120} = \frac{1}{2} \]

### Final Answer:
The probability that the student passes the exam is \( \frac{1}{2} \).

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