Answer:
The probability that at least two stocks will have a return of more than 12% is 0.1810.
Step-by-step explanation:
Let <em>X</em> = rate of return on stocks.
The random variable <em>X</em> follows a Normal distribution, N (9, 3²).
Compute the probability that a stock has rate of return more than 12% as follows:
![P(X\geq 12)=1-P(X](https://tex.z-dn.net/?f=P%28X%5Cgeq%2012%29%3D1-P%28X%3C12%29%5C%5C%3D1-P%28%5Cfrac%7BX-%5Cmu%7D%7B%5Csigma%7D%3C%5Cfrac%7B12-9%7D%7B3%7D%20%20%29%5C%5C%3D1-P%28Z%3C1%29%5C%5C%3D1-0.8413%5C%5C%3D0.1587)
**Use the <em>z</em> table for the probability.
The probability of a stock having rate of return more than 12% is 0.1587.
Now define a random variable <em>Y</em> as the number of stocks that has rate of return more than 12%.
The sample size of stocks selected is, <em>n</em> = 5.
The random variable <em>Y </em>follows a Binomial distribution.
The probability of a Binomial distribution is:
![P(X=x)={n\choose x}p^{x}(1-p)^{n-x};\ x=0, 1, 2, ...](https://tex.z-dn.net/?f=P%28X%3Dx%29%3D%7Bn%5Cchoose%20x%7Dp%5E%7Bx%7D%281-p%29%5E%7Bn-x%7D%3B%5C%20x%3D0%2C%201%2C%202%2C%20...)
Compute the value of P (X ≥ 2) as follows:
P (X ≥ 2) = 1 - P (X < 2)
= 1 - P (X = 0) - P (X = 1)
![=1-{5\choose 0}(0.1587)^{0}(1-0.1587)^{5-0}-{5\choose 1}(0.1587)^{1}(1-0.1587)^{5-1}\\=1-0.4215-0.3975\\=0.1810](https://tex.z-dn.net/?f=%3D1-%7B5%5Cchoose%200%7D%280.1587%29%5E%7B0%7D%281-0.1587%29%5E%7B5-0%7D-%7B5%5Cchoose%201%7D%280.1587%29%5E%7B1%7D%281-0.1587%29%5E%7B5-1%7D%5C%5C%3D1-0.4215-0.3975%5C%5C%3D0.1810)
Thus, the probability that at least two stocks will have a return of more than 12% is 0.1810.