Bacteria as pets-seriously?
Probability
Level
3
X was bored one day and they decided to raise bacteria as pets. X started with 1 bacterium and in every second, each(in this case,1)bacterium turns into 0,1,2 and 3 bacteria, with equal probability of each outcome.What is the probability that the colony will eventually die out?
0
sqrt(2)-1
sqrt(4)+3
1/4
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1 solution
I used a more algebraic approach to the problem, which I saw in a book. Let the probability of the colony of bacteria dying out be x, and the number of bacteria in a colony be n. Then the probability of eventually dying out is x^n, as each bacterium can be considered as a colony. Comparing the number of bacteria before and after the first second, we have: x=1/4 times 1+1/4 times x^2+1/4 times x^3 The equation will become: x^3+x^2-3x+1=0 Factorizing this equation, we have: (x-1)(x^2+2x-1)=0 Hence the roots are:(1),(-1-sqrt(2)),(sqrt(2)-1). Straight away, we can exclude 1 and (-1-sqrt(2)), because it would mean all would eventually die out, and also the probability cannot be a negative answer. Hence the only solution is sqrt(2)-1.