Jack says he can predict and wants to test out which barrel has water in it and which does not. The test consists of giving Jack 10 times 2 barrels which are completely identical, other than the fact that one of them has water in it.

1. Let X be the number of correct choices. If jack senses there is oil in a barrel with a 0.8 probability what is the distribution of X, the expected value (E[X]) and the variance (var[X])?
2. If jack predicts there is water with a probability of 0.8, what is the probability that Jack has chosen the right barrel at least 8 times?
3. What would be the minimum probability that Jack had 8 barrels guessed correctly if he would have secretly lied and just randomly chose the barrels?