Question

1. Consider an urn with 4 blue balls, 6 red balls, and 3 yellow balls. Suppose we draw 4 balls at random. (a) How many elements are in the sample space? (b) What is the probaiblity that we draw 4 red balls? (c) What is the probability that we draw 2 red balls and 2 blue balls? (d) What is the probability that we draw either 3 blue and 1 yellow ball or 1 blue and 3 yellow balls? 2. Consider a diffusion model where the probability a molecule exits a cell in a one-second interval is 0.3 while the probability it re-enters the cell in a one-second interval is 0.2. Let pe be the probabilty the molecule is inside the cell at t seconds for t=0,1,2,.... (a) If the molecule starts inside, i.e. Po = 1, compute the probability the molecule is inside at t +1, P+1, in terms of pe (b) Find P2, the probability the molecule is inside at t = 2. (c) Find the probability the molecule is outside at t = 2. (d) Compute the equilibrium of the system, which is the probability the molecule is inside "in the long run". 3. Consider a self-pollinating species where AA and Aa plants are green and aa plants are purple. Suppose we have an Aa plant and are observing its generational offspring. Let pebe the probability that a plant from the t-th generation is Aa. (a) Compute po, the probability the original plant is Aa. (b) Compute p, the probability that a plant from the first generation of offspring is Aa (c) Compute pa (d) Computer

Answer 1

1. a) There are a blue balls, & red talis 3 yellow boks . :. Total elements or balls in the затра зраас ога (4+6+3) = 12, b) Probability that we draw 4 red balls Баѕ is is 16 (f) » 716 As we can chose a red ball from 6 ла 8. bo8 "м (4) - 6e4 д оv4 5 , And all possible cases are (13) = 139 2) Probability that we draw a redball and 2 bine ball is, (3) (3). 15х6 - 1 15 145 as from 6 red ball we can chose a red ball in (6) = 66₂ ways and we can chose a blue ball from a blue ball in (4) - 4с, ну 5. d) эрим бил 8 yellow Y Abus емі 3 yellow Mo Coм ље сло40 Ал м (4)(0) [3) Я у 6.
9
so is, required probability (3) (3) + 1303) 16 4x314x], - 15