Chapter PIV, Problem 16RE

(a)

To determine

To find: the probability for the market would rise for the three consecutive years.

Answer to Problem 16RE

0.39

Explanation of Solution

The probability that stock prices have increased is 0.73

From year to year, the market output is independent.

The possibility that stock prices will have increased for a consecutive year is,

  $\begin{array}{c}P\left(\text{rise}\right)=\left(0.73\right)\left(0.73\right)\left(0.73\right)\\ =0.389017\\ =0.39\end{array}$

For three consecutive years, there is a 39 percent risk that stock prices have increased.

(b)

To determine

To find: the probability that the market would rise three years out of the next 5.

Answer to Problem 16RE

0.28

Explanation of Solution

Given:

  $\begin{array}{c}n=5\\ x=3\\ p=0.73\end{array}$

Formula used:

  $P\left(X=x\right)=C{}_x{\left(p\right)}^x{\left(1-p\right)}^{n-x}$

Calculation:

Let X be the number of years in which the stock market will increase.

X is binom (5, 0.73)

The probability of an increase in the demand for 3 years out of the next 5 years is,

  $\begin{array}{c}P\left(X=x\right)=C{}_x{\left(p\right)}^x{\left(1-p\right)}^{n-x}\\ P\left(X=3\right)=C{}_3{\left(0.73\right)}^3{\left(1-0.73\right)}^{5-3}\\ =10\times 0.3890\times 0.0729\left(C{}_x=\frac{n!}{\left(n-x\right)!x!}\right)\\ =0.283593\\ =0.28\end{array}$

Therefore, the required probability is 0.28.

(c)

To determine

To find: the probability that the market would fall during at minimum one of the next five years.

Explanation of Solution

Given:

  $\begin{array}{c}n=5\\ x=3\\ p=0.73\end{array}$

Calculation:

The probability of the demand increasing for at least 1 of the next 5 years is,

  $\begin{array}{c}P\left(X\ge 1\right)=1-P\left(X=0\right)\\ =1-C{}_3{\left(0.73\right)}^0{\left(0.27\right)}^5\\ =1-\left(1\times 1\times 0.0014\right)\\ =1-0.0014\\ =0.9986\end{array}$

There is a 99.86 percent probability that at least 1 of the next 5 years will increase the demand.

(d)

To determine

To find: the possibility that the demand will grow for the next decade for the majority of the year.

Answer to Problem 16RE

0.90

Explanation of Solution

Given:

  $\begin{array}{c}n=10\\ p=0.73\end{array}$

Calculation:

The target is to recognise the possibility that in the next decade the demand will increase over most years.

10 years means a decade.

The majority over the next decade means the demand will grow in more than half of the years.

So, find P(X > 5)

Here X is binom (10,0.73)

In the next decade, the probability that the demand will grow for most years is,

  $\begin{array}{c}P\left(X>5\right)=1-P\left(X\le 5\right)\\ =1-{\displaystyle \sum _{x=0}^{5}C{}_x{\left(0.73\right)}^0{\left(0.27\right)}^{10-x}}\\ =1-\left[\begin{array}{l}C{}_0{\left(0.73\right)}^0{\left(0.27\right)}^{10-0}+C{}_1{\left(0.73\right)}^1{\left(0.27\right)}^{10-1}\\ +C{}_2{\left(0.73\right)}^2{\left(0.27\right)}^{10-2}+C{}_3{\left(0.73\right)}^3{\left(0.27\right)}^{10-3}\\ +C{}_4{\left(0.73\right)}^4{\left(0.27\right)}^{10-4}+C{}_5{\left(0.73\right)}^5{\left(0.27\right)}^{10-5}\end{array}\right]\\ =1-\left[\begin{array}{l}0.0000+0.0001+0.0007\\ +0.0049+0.0231+0.0750\end{array}\right]\\ =1-0.1037\\ =0.8963\\ =0.90\end{array}$

There is a 90 percent probability that in the next decade the demand will grow across most years.