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4. A biologist is cultivating a kind of super-bacteria in his laboratory. In a successful experiment,...

4. A biologist is cultivating a kind of super-bacteria in his laboratory. In a successful experiment, the number of super-bacteria will be doubled. However, the number of super-bacteria will stay the same if the experiment fails. (a) Suppose now there are exactly 1 million super-bacteria 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 super-bacteria 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 super-bacteria, and he has exactly 1 million super- bacteria to begin with. In his laboratory, the number of super-bacteria will be doubled if the experiment is successful. However, the number of super-bacteria 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.

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