Question: a biologist is cultivating a kind of superbacteria in his...
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A biologist is cultivating a kind of superbacteria in his laboratory. In a successful experiment, the number of superbacteria will be doubled. However, the number of superbacteria will stay the same if the experiment fails.

(a) Suppose now there are exactly 1 million superbacteria and the experiment has a 50 % chance of failing. Also suppose that the outcomes of individual experiments are independent of each other. What is the probability that there are exactly 16 million superbacteria after doing 10 experiments? Also give a general expression for the probability that there are exactly b million bacteria after a experiments.

(b) Another biologist is also cultivating such superbacteria, and he has exactly 1 million super bacteria to begin with. In his laboratory, the number of superbacteria will be doubled if the experiment is successful. However, the number of superbacteria will decrease by a factor of two if the experiment fails. The probability of a successful experiment is 1/2, and that of a failed experiment is also 1/2. Suppose that the outcomes of the individual experiments are independent of each other. What is the probability that there are exactly 4 million bacteria after 10 experiments? Also give a general expression for the probability that there are exactly b million bacteria after a experiments.