A colony that booms or shrinks each generation

Question

A colony has 50 members. Each generation, independently, its size is multiplied by 3 with probability 	frac13 (a boom) or by 0.6 with probability 	frac23 (a decline). It runs for two generations. What size should you expect after two generations?

Accepted Answer

Each generation multiplies the colony size by a random factor, and generations are independent. Condition on the size before each generation with the tower rule. Step 1, the average factor per generation. E[	ext{factor}] = 	frac13 	imes 3 + 	frac23 	imes 0.6 = 1 + 0.4 = 1.4. Step 2, the recursion. Let St be the size after t generations. Given S{t-1}, the next generation multiplies it by an independent factor averaging 1.4, so E[St] = 1.4 \, E[S{t-1}]. Step 3, iterate from S0 = 50 over two generations: E[S2] = 50 	imes 1.4^2 = 50 	imes 1.96 = 98. So you expect a colony size of 98. Decli…

A colony that booms or shrinks each generation

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