Buying stock $(5.3,6.1)$You purchase a hot stock for $1000$. The stock either gains $30\%$or loses $25\%$each day, each with probability $0.5$. Its returns on consecutive days are independent of each other. You plan to sell the stock after two days.

(a) What are the possible values of the stock after two days, and what is the probability for each value? What is the probability that the stock is worth more after two days than the $1000$you paid for it?

(b) What is the mean value of the stock after two days?

(Comment: You see that these two criteria give different answers to the question "Should I invest?")

Hot stock price $=1000$

Gain each day$=30\%$

Loss each day$=25\%$

Probability$=0.5$

Find the worth of the stock after the first day:

Gain: $\$1000+\$1000\times 30\%=\$1300$

Loss:localid="1649856969788" $\$1000-\$1000\times 25\%=\$750$

Find the worth of the stock after the second day:

2 Gains:$\$1300+\$1300\times 30\%=\$1690$

Gain+Loss (or Loss+Gain): $\$1300-\$1300\times \$25\%=\$975$

2 Losses: $\$750-\$750\times 25\%=\$562.50$

The probability of each outcome is equal (note: first a gain and then a loss is the same as obtaining first a loss and then a gain):

localid="1649856974109" $$\begin{gathered} P(\$1690)=\left(\frac{1}{4}\right) \\ =0.25 \\ P(\$975)=\left(\frac{2}{4}\right) \\ =\$0.5 \\ P(\$562.50)=\left(\frac{1}{4}\right) \\ =0.15\$ \end{gathered}$$

Hot stock price$=1000$

Gain each day$=30\%$

Loss each day$=25\%$

Probability$=0.5$

Find the worth of the stock after the first day:

Gain: $\$1000+\$1000\times 30\%=\$1300$

Loss: $\$1000-\$1000\times 25\%=\$750$

Find the worth of the stock after the second day:

2 Gains: $\$1300+\$1300\times 30\%=\$1690$

Gain+Loss (or Loss+Gain): $\$1300-\$1300\times 25\%=\$975$

2 Losses: $\$750-\$750\times 25\%=\$562.50$

It is equally likely that you will obtain either a gain or a loss (note: obtaining a gain followed by a loss is the same as obtaining a loss followed by a gain):

localid="1649856982682" $$\begin{gathered} P\left(\mathrm{\$}1690\right)=\frac{1}{4} \\ =0.25 \\ P\left(\mathrm{\$}975\right)=\frac{2}{4} \\ =0.5 \\ P\left(\mathrm{\$}562.50\right)=\frac{1}{4} \\ =0.25 \end{gathered}$$

Each possibility is multiplied by its probability to arrive at the expected value:

$$\begin{gathered} \mu =\sum xP\left(x\right) \\ =\mathrm{\$}1690\times 0.25+\mathrm{\$}975\times 0.5+\mathrm{\$}562.50\times 0.25 \\ =\mathrm{\$}1050.625 \end{gathered}$$