Question

An investor holds two stocks, each of which can rise (R), remain unchanged (U), or decline (D) on any particular day. Assume that for the first stock (on a particular day)

P(R) = 0.2 , P(U) = 0.6 , P(D ) = 0.2

and that for the second stock (on a particular day)

P(R) = 0.3 , P(U) = 0.4 , P(D ) = 0.3

Assuming that these stocks move independently, find the probability that both stocks decline; the probability that exactly one stock rises; the probability that exactly one stock is unchanged; the probability that both stocks rise. (Round your answers to 2 decimal places.)

Both decline =

Exactly one rises =

Exactly one unchanged =

Both rise =

Answer 1

Answer:

1. 0.06

2. 0.38

3. 0.52

4. 0.06

Explanation:

1) Both decline = P1(D) × P2(D)

                         = 0.2 × 0.3

                         = 0.06

(stock 1 declines and stock 2 declines)

2) Exactly one rises:

= P1(R) × [P2(U) + P2(D)] + P2(R) × [P1(U) + P1(D)]

= 0.2 × [0.4+ 0.3] + 0.3 × [0.6 + 0.2]

= 0.14 + 0.24

= 0.38

(stock 1 rises and {stock 2 declines or remains unchanged})

or (stock 2 rises and {stock 1 declines or remains unchanged})

3) Exactly one unchanged:

= P1(U) × [P2(R) + P2(D)] + P2(U) × [P1(R) + P1(D)]

= 0.6 × [0.3 + 0.3] + 0.4 × [0.2 + 0.2]

= 0.36 + 0.16

= 0.52

4) Both rise = P1(R) × P2(R)

                   = 0.2 × 0.3

                   = 0.06

Answer 2

The probability that both decline is 0.06, the probability that exactly one rises will be 0.38; for exactly one being unchanged will be 0.52 and that both rise will be 0.06.

How to calculate probability?

Probability is referred to as the chances of occurrence of a product in an event or series of events.

From the given information, considering price of two stocks as A and B. The formula for calculation of probability when prices of both stocks should decline will be as,

$\rm Probability\ of\ Decline\ in\ A\ and\ B = PA(D)\ x\ PB(D)\\\\\rm Probability\ of\ Decline\ in\ A\ and\ B =0.2\ x\ 0.3\\\\\rm Probability\ of\ Decline\ in\ A\ and\ B =0.06$

Now probability of exactly one price rises will be,

$\rm Probability\ that\ one\ rises= PA(U)\ x\ [PB(R) + PB(D)] + PB(U)]\ x\ [PA(R)\ +\ PA(D)]\\\\\rm Probability\ that\ one\ rises=0.2\ x [0.4+ 0.3] + 0.3\ x\ [0.6 + 0.2]\\\\\rm Probability\ that\ one\ rises= 0.38$

Now for exactly one price remaining unchanged,

$\rm Probabiility\ for\ One\ Unchanged = PA(U)\ x\ [PB(R) + PB(D)] + PB(U)\ x\ [PA(R) + P1(D)]\\\\\rm Probabiility\ for\ One\ Unchanged = 0.6\ x\ [0.3 + 0.3] + 0.4\ x\ [0.2 + 0.2]\\\\\rm Probabiility\ for\ One\ Unchanged = 0.52$

Probability that both the prices rise will be,

$\rm Probability\ that\ Both\ Rise\ = PA(R) × PB(R)\\\\\rm Probability\ that\ Both\ Rise\ =0.2\ x\ 0.3\\\\\rm Probability\ that\ Both\ Rise\ = 0.06$

Hence, the different probabilities related to price fluctuations of stocks held by an investor are computed and calculated as in the steps above.

Learn more about probability here:

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