<p>A binary number is made up of 8 digits. Suppose that the probability of an incorrect digit appearing is \(p\) and that the errors in different digits are independent of each other. Then find the probability of forming an incorrect number.</p>
Step-by-Step Solution
Key Concept: An incorrect number means at least one digit is wrong. Use complement: P(at least one error) = 1 - P(no errors). Since errors are independent, P(no errors) = product of individual probabilities of being correct.
<p><strong>Step 1:</strong> Define the complement event. A number is incorrect if at least one of its 8 digits is incorrect. The complement is that all 8 digits are correct.</p><p><strong>Step 2:</strong> Find probability that a single digit is correct: P(digit correct) = 1 - p</p><p><strong>Step 3:</strong> Since errors in different digits are independent, use the multiplication rule: P(all 8 digits correct) = (1-p) × (1-p) × ... × (1-p) = (1-p)^8</p><p><strong>Step 4:</strong> Apply complement rule: P(at least one error) = 1 - P(no errors) = 1 - (1-p)^8</p><p>∴ Answer: 1-(1-p)^8</p>
Correct Answer: 1-(1-p)^8