Three defective bulbs got mixed up with seven good ones. If two bulbs are selected at random, what is the probability that both are good?

Answer Details

We can solve this problem using the concept of probability. Let's begin by finding the total number of ways to select two bulbs out of ten. We can do this using the combination formula: $${10 \choose 2} = \frac{10!}{2!(10-2)!} = 45$$ So there are 45 ways to select two bulbs out of ten. Now let's find the number of ways to select two good bulbs out of the seven good ones. We can use the combination formula again: $${7 \choose 2} = \frac{7!}{2!(7-2)!} = 21$$ So there are 21 ways to select two good bulbs out of seven. Finally, we can find the probability of selecting two good bulbs by dividing the number of ways to select two good bulbs by the total number of ways to select two bulbs: $$P(\text{both bulbs are good}) = \frac{21}{45} = \frac{7}{15}$$ Therefore, the answer is, the probability that both bulbs are good is $\frac{7}{15}$.