A box contains 5 defective and 10 non-defective lamps. Eight lamps are drawn at random insuccession without replacement. What is the probability that the 8th lamp is the 5th defective?
Question
A box contains 5 defective and 10 non-defective lamps. Eight lamps are drawn at random insuccession without replacement. What is the probability that the 8th lamp is the 5th defective?
Solution
To solve this problem, we need to understand that we are looking for the probability that the 5th defective lamp is the 8th lamp drawn. This means that in the first 7 draws, we must have drawn 4 defective lamps and 3 non-defective lamps.
Step 1: Calculate the probability of drawing 4 defective lamps and 3 non-defective lamps in the first 7 draws.
The total number of ways to draw 4 defective lamps from 5 is given by the combination 5C4. Similarly, the total number of ways to draw 3 non-defective lamps from 10 is given by the combination 10C3.
The total number of ways to draw 7 lamps from 15 (5 defective + 10 non-defective) is given by the combination 15C7.
So, the probability of drawing 4 defective lamps and 3 non-defective lamps in the first 7 draws is (5C4 * 10C3) / 15C7.
Step 2: Calculate the probability that the 8th lamp drawn is defective.
At this point, there is 1 defective lamp and 7 non-defective lamps left. So, the probability that the 8th lamp drawn is defective is 1/8.
Step 3: Multiply the results from step 1 and step 2 to get the final probability.
So, the probability that the 8th lamp is the 5th defective is (5C4 * 10C3) / 15C7 * 1/8.
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