Probability
Conditional Probability
Grade 12
Question:
<p>A student can solve 2 out of 4 problems of mathematics, 3 out of 5 problem of physics, and 4 out of 5 problems of chemistry. There are equal number of books of math, physics, and chemistry in his shelf. He selects one book randomly and attempts 10 problems from it. If he solves the first problem, then the probability that he will be able to solve the second problem is</p>
<p>(1) 2/3</p>
<p>(2) 25/38</p>
<p>(3) 13/21</p>
<p>(4) 14/23</p>
Step-by-Step Solution
Key Concept: Use Bayes' theorem to find the probability of solving the 2nd problem given the 1st was solved. The posterior probabilities of which book was selected must be updated based on the evidence of solving the first problem.
<p><strong>Step 1:</strong> Identify the solving probabilities for each subject:</p><ul><li>Mathematics: P(M) = 2/4 = 1/2</li><li>Physics: P(P) = 3/5</li><li>Chemistry: P(C) = 4/5</li></ul><p><strong>Step 2:</strong> Since he solved the first problem, use Bayes' theorem to find the posterior probability of selecting each book:</p><p>P(Book | 1st solved) ∝ P(1st solved | Book) × P(Book)</p><p>P(Math | 1st solved) = (1/2 × 1/3)/(1/2 × 1/3 + 3/5 × 1/3 + 4/5 × 1/3) = (1/6)/(1/6 + 1/5 + 4/15)</p><p>= (1/6)/(5/30 + 6/30 + 8/30) = (5/30)/(19/30) = 5/19</p><p>P(Physics | 1st solved) = (3/5 × 1/3)/(19/30) = (1/5)/(19/30) = 6/19</p><p>P(Chemistry | 1st solved) = (4/5 × 1/3)/(19/30) = (4/15)/(19/30) = 8/19</p><p><strong>Step 3:</strong> Calculate P(2nd solved | 1st solved) using total probability:</p><p>P(2nd | 1st solved) = (5/19)(1/2) + (6/19)(3/5) + (8/19)(4/5)</p><p>= (1/19)[5/2 + 18/5 + 32/5]</p><p>= (1/19)[25/10 + 36/10 + 64/10]</p><p>= (1/19)(125/10) = 125/190 = 25/38</p><p>∴ Answer: <strong>25/38</strong></p>
Correct Answer: 2