please do not use Excel

Hypergeometric distribution 5.20 There are 500 deer in a wildlife preserve. A sample of 50 deer are caught and tagged and returned to the population. Suppose that 20 deer are caught. (a) Find the mean and standard deviation of the number of tagged deer in the sample. (b) Find the probability that the sample contains at least three tagged deer. 5.21 The Lady Tasting Tea. This is one of the most famous experiments in the founding history of statistics. In his 1935 book The Design of Experiments (1935), Sir Ronald A. Fisher writes, A Lady declares that by tasting a cup of tea made with milk she can discriminate whether the milk or the tea infusion was first added to the cup. We will consider the problem of designing an experiment by means of which this assertion can be tested . . . Our experiment consists in mixing eight cups of tea, four in one way and four in the other, and presenting them to the subject for judgment in a random order. . . . Her task is to divide the 8 cups into two sets of 4, agreeing, if possible, with the treatments received. Consider such an experiment. Four cups are poured milk first and four cups are poured tea first and presented to a friend for tasting. Let X be the number of milk-first cups that your friend correctly identifies as milk-first. (a) Identify the distribution of X. (b) Find P(X = k) for k = 0, 1, 2, 3, 4. (c) If in reality your friend had no ability to discriminate and actually guessed, what is the probability they would correctly identify all four cups correctly? 5.22 In a town of 20,000, there are 12,000 voters, of whom 6000 are registered democrats and 5000 are registered republicans. An exit poll is taken of 200 voters. Assume all registered voters actually voted. Use R to find: (i) the mean and standard deviation of the number of democrats in the sample; (ii) the probability that more than half the sample are republicans