Amodel for the movement of a stock supposes that if the present price of the stock is s, then, after one period, it will be either us with probability p or ds with probability 1 − p. assuming that suc- cessive movements are independent, approximate the probability that the stock’s price will be up at least 30 percent after the next 1000 periods if u = 1.012,d = 0.990, and p = .52.

Step-by-step explanation:

Present price of stock is s, if x is the times that the stock increases (by u) among the 1000 periods we can calculate the price after 1000 periods as follows:

note that if the price increase n periods, so it should decreases in the other 1000-x periods.

We need that the price be 30% higher or 1.3 times the initial price. That requirement is expressed as follows:

applying properties of exponents and reorganizing.

solving for x

Using the values given d= 0.99 and u=1.012

So, we need at least 470 increase periods

Now we can calculate the probability of have 470 increase periods or more, taking into account that the distribution of x is binomial, the period can be and increase period with 0.52 of probability and we have 1000 total trials.

The formula for the binomial cumulative probability function is

With this formula we can calculate the probability of obtain x successful trials or less (). There, p is the probability of success (probability of increase period in this case), x is the maximum number of successful trials, n the number of total trials.  We should calculate F with x=470, p=0.52 and n= 1000. The result is , but we need so we can calculate that probability as

In order to calculate the cumulative function we can use excel function BINOM.DIST

BINOM.DIST(x,trials,probability,cumulative)

BINOM.DIST(470,1000,0.52,TRUE)

We obtain that  , so  

45

step-by-step explanation:

∠acb = 8°

step-by-step explanation:

∠xba lies on the circle and is half the measure of the intercepted arc ax

hence

measure of arc ax = 2 × 42° = 84°

∠acb is outside the circle and is

∠acb = (arcab - arcax ), hence

∠acb = 0.5 (100 - 84) = 0.5 × 16° = 8°