Question:

A packet contains 10 distinguishable firecrackers out of which 4 are defective. If three firecrackers are drawn at random (without replacement) from the packet, then what is the probability that all three firecrackers are defective? Explain in detail.

Answer:

To solve this problem, we need to calculate the probability of drawing all three defective firecrackers from the packet. Let's break down the solution into the following steps:

Step 1: Determine the total number of ways to choose three firecrackers:

Since we are drawing without replacement, the total number of ways to choose three firecrackers from a packet of 10 is given by the combination formula:

C(10, 3) = 10! / (3! * (10 - 3)!) = 120.

Therefore, there are 120 possible ways to choose three firecrackers from the packet.

Step 2: Determine the number of ways to choose three defective firecrackers:

Out of the 10 firecrackers, 4 are defective. The number of ways to choose three defective firecrackers from the packet is given by the combination formula:

C(4, 3) = 4! / (3! * (4 - 3)!) = 4.

Therefore, there are 4 possible ways to choose three defective firecrackers from the packet.

Step 3: Calculate the probability:

The probability of drawing three defective firecrackers can be calculated by dividing the number of ways to choose three defective firecrackers by the total number of ways to choose three firecrackers:

P(3 defective firecrackers) = (Number of ways to choose 3 defective firecrackers) / (Number of ways to choose 3 firecrackers)

P(3 defective firecrackers) = 4 / 120

Simplifying, P(3 defective firecrackers) = 1 / 30.

Therefore, the probability of drawing all three defective firecrackers from the packet is 1/30 or approximately 0.0333.

Conclusion:

The probability that all three firecrackers drawn at random from the packet are defective is 1/30 or approximately 0.0333.