Have no ideas how to start. Any thoughts?
Ivan Nikiforovich invests his savings in a mutual investment fund. The mutual fund yield is a random variable uniformly distributed over the interval [-10,20]% per annum. Until his retirement, Ivan Nikiforovich has 10 more years left. What is the probability of his savings during this time to grow at least twice?
My idea is to calculate the average growth rate which lead to the double of the investment after 10 years. The equation is $(1+x)^{10}=2\Rightarrow x=2^{\frac1{10}}-1=0.0717734625362931$. With an mutual fund yield of at least $7.17735\%$ we can ensure that investment is at least twice after 10 years.
Next we have to calculate the probability that the mutual fund yield is greater than $7.17735\%$. Here we use the uniform distribution. $$P(X\geq 7.17735\%)=1-P(X\leq 7.17735\%)=1-\frac{0.0717735-(-0.1)}{0.2-(-0.1)}=0.4274217$$
Same approach with a slightly different perspective. We can just re-arrange the inequality. Basically the asked probability is $$P\left(\left(1+X \right)^{10}\geq 2\right),$$
where $X\sim U(-0.1,0.2)$
After some transformations we get
$$P\left(X \geq 2^{\frac1{10}}-1\right)$$
Thus Ivan Nikiforovich has a probability of $\color{blue}{42.74\%}$ that his savings grow at least twice in ten years.