Probability
Classical Probability
Grade 12
Question:
<p>A real estate man has eight master keys to open several new homes. Only one master key will open any given house. If 40% of these homes are usually left unlocked, what is the probability that the real estate man can get into a specific home if he selects three master keys at random before leaving the office?</p>
Step-by-Step Solution
Key Concept: The home can be entered either if it's unlocked (40% chance) OR if it's locked but one of the 3 randomly selected keys opens it. Use the complement principle: P(can enter) = P(unlocked) + P(locked AND correct key selected).
<p><strong>Step 1:</strong> Identify the two independent ways to enter the home:</p><ul><li>Home is unlocked (probability = 0.40)</li><li>Home is locked BUT the correct key is among the 3 selected</li></ul><p><strong>Step 2:</strong> Calculate P(can open | home is locked):</p><p>P(correct key among 3 selected) = 3/8 = 0.375</p><p>P(locked) = 1 - 0.40 = 0.60</p><p><strong>Step 3:</strong> Apply the addition rule for mutually exclusive scenarios:</p><p>P(can enter) = P(unlocked) + P(locked AND correct key selected)</p><p>P(can enter) = 0.40 + (0.60 × 3/8)</p><p>P(can enter) = 0.40 + 0.225 = 0.625</p><p>∴ Answer: <strong>0.625 or 5/8 or 62.5%</strong></p>
Correct Answer: 0