Probability
Total Probability Theorem
Grade 12
Question:
<p>Two-thirds of the students in a class are boys and the rest are girls. It is known that probability of a girl getting a first class is 0.25 and that of a boy is 0.28. Then find the probability that a student chosen at random will get a first class.</p>
Step-by-Step Solution
Key Concept: Use the law of total probability by conditioning on gender: P(First Class) = P(First Class|Boy)·P(Boy) + P(First Class|Girl)·P(Girl). The given probabilities are conditional probabilities for each gender.
<p><strong>Step 1:</strong> Identify the given information:</p><ul><li>P(Boy) = 2/3, so P(Girl) = 1/3</li><li>P(First Class | Girl) = 0.25</li><li>P(First Class | Boy) = 0.28</li></ul><p><strong>Step 2:</strong> Apply the law of total probability:</p><p>P(First Class) = P(First Class | Boy) × P(Boy) + P(First Class | Girl) × P(Girl)</p><p><strong>Step 3:</strong> Substitute the values:</p><p>P(First Class) = 0.28 × (2/3) + 0.25 × (1/3)</p><p>P(First Class) = 0.56/3 + 0.25/3</p><p>P(First Class) = 0.81/3 = 0.27</p><p>∴ Answer: <strong>0.27 or 27/100</strong></p>
Correct Answer: 0