Probability
Conditional Probability and Independence
Grade 12
Question:
<p>Four persons independently solve a problem with probabilities \(\dfrac{1}{2}, \dfrac{3}{4}, \dfrac{1}{4}, \dfrac{1}{8}\). The probability that the problem is solved correctly by <strong>at least one</strong> of them is <em>[JEE Advanced 2013]</em></p>
The probability that the problem is solved correctly by at least one of them is 1 - (1/2) * (1 - 3/4) * (1 - 1/4) * (1 - 1/8)
The probability that the problem is solved correctly by at least one of them is 1 - (1/2) * (1 - 3/4) * (1 - 1/4) * (1/8)
The probability that the problem is solved correctly by at least one of them is 1 - (1/2) * (3/4) * (1 - 1/4) * (1 - 1/8)
The probability that the problem is solved correctly by at least one of them is 1 - (1/2) * (3/4) * (1/4) * (1 - 1/8)
Step-by-Step Solution
Key Concept: P(at least one solves) = 1 - P(none solve) = 1 - (1-1/2)(1-3/4)(1-1/4)(1-1/8).
<p>$P(\text{none solve}) = \left(\frac{1}{2}\right)\left(\frac{1}{4}\right)\left(\frac{3}{4}\right)\left(\frac{7}{8}\right) = \frac{21}{256}$.</p><p>$P(\text{at least one}) = 1 - \frac{21}{256} = \frac{235}{256}$</p>
Correct Answer: A