A research scientist is monitoring a chemical synthesis reaction in which 20% of batches are historically found to contain a specific precipitate. Each batch is independent of all others.
Find the probability that, in a random sample of 15 batches, the number of batches containing the precipitate is
exactly 4,
at most 3.
After adjusting the reaction temperature, the probability that each batch contains the precipitate, independently of all others, is changed to 0.35.
The scientist then performs a further 80 batches.
Calculate the mean and variance for the number of these 80 batches that contain the precipitate.
Using a suitable approximation, calculate the probability that the precipitate is present in more than 32 of these batches.
