Here's the problem I have:

The probability that an apple is red is 80%. The probability that a red apple is rotten is 10%. If 10 apples are picked at random, what are the odds that half are red and rotten?

Approach 1:

The odds of an apple being both red and rotten is 8%. Therefore, we have a binomial distribution with 10 trials, 5 expected successes, and probability of success 0.08.

This gives us a probability of 0.054%.

Approach 2:

We would expect, on average, to get 8 red apples when we pick 10. This allows us to skip the 80% and compute a binomial distribution with the following: 8 trials, 5 expected successes, and probability 0.1.

Here, we get a probability of 0.041%

I believe the right answer comes from Approach 1, but I'm not sure. Which one is correct and why do we get different numbers?