In a discount store, 12% of the 100 cups in stock have some unnoticeable defect. What is the probability of buying 12 cups that don’t have any defect? And what is the probability of getting all 12 cups having a defect?
The Correct Answer and Explanation is:
Step 1: Define the probabilities
- Probability of selecting a non-defective cup = 88/100 = 0.88
- Probability of selecting a defective cup = 12/100 = 0.12
We want to compute:
- The probability that all 12 selected cups are non-defective: P(All 12 non-defective)=0.8812≈0.219P(\text{All 12 non-defective}) = 0.88^{12} \approx 0.219P(All 12 non-defective)=0.8812≈0.219
- The probability that all 12 selected cups are defective: P(All 12 defective)=0.1212≈5.15×10−13P(\text{All 12 defective}) = 0.12^{12} \approx 5.15 \times 10^{-13}P(All 12 defective)=0.1212≈5.15×10−13
Explanation
This is a classic case of independent Bernoulli trials, where each trial (or selection of a cup) results in one of two outcomes: defective or non-defective. Since the selections are assumed to be independent and randomly chosen, we can multiply the probabilities for each selection to find the probability of a sequence of outcomes.
When calculating the probability that all 12 cups are non-defective, we raise the probability of selecting a single non-defective cup (0.88) to the 12th power:0.8812≈0.2190.88^{12} \approx 0.2190.8812≈0.219
This means there’s approximately a 21.9% chance that all 12 cups you randomly pick from the store will be non-defective — fairly likely, but not guaranteed.
On the other hand, the chance of picking all defective cups is extremely small. Since only 12 out of 100 cups are defective, the probability of selecting one defective cup is 0.12. Raising this to the 12th power gives:0.1212≈0.0000000000005150.12^{12} \approx 0.0000000000005150.1212≈0.000000000000515
This probability is virtually zero, meaning it is almost impossible to randomly select 12 defective cups in a row.
In real-world terms, this reflects the rarity of defective items dominating a random selection if their overall frequency is low. It also shows that even a seemingly decent defect rate (12%) leads to a high likelihood of getting some good items when selecting in small quantities — but getting only bad ones is highly unlikely.Tools
