The increase or decrease in the price of a stock between the beginning and the end of a trading day is assumed to be an equally likely random event. What is the probability that a stock will show a decrease in its closing price on eleven consecutive days?
(Round to four decimal places)
So I was given the formula P(x)= n! / x! (n-x)! π^x (1-π)^n-x
I got these values
n=11
x = 11
π = 0.5 since P(increase) = P(decrease) = 1/2
I used the formula above but got the incorrect answer. Can i get some help with this problem? Thanks.
The correct answer is 0.0098.
Using the formula you provided, the answer would be:
P(x) = 11! / 11! (11-11)! (0.5)^11 (0.5)^(11-11) = 0.0098
To solve this problem, you need to calculate the probability of a stock showing a decrease in its closing price on eleven consecutive days. You are given the formula P(x) = n! / x! (n-x)! π^x (1-π)^n-x, where:
P(x) represents the probability of obtaining exactly x successes in n trials.
n represents the total number of trials.
x represents the number of successful trials.
π represents the probability of success in a single trial.
Given that n = 11, x = 11, and π = 0.5 (since both increase and decrease have an equal chance of occurring), let's calculate the probability step by step:
1. Calculate the factorials:
n! = 11! = (11)(10)(9)(8)(7)(6)(5)(4)(3)(2)(1) = 39,916,800
x! = 11! = (11)(10)(9)(8)(7)(6)(5)(4)(3)(2)(1) = 39,916,800
(n - x)! = (11 - 11)! = 0! = 1
2. Determine π^x:
π^x = 0.5^11 = 0.00048828125 (to 11 decimal places)
3. Determine (1 - π)^(n - x):
(1 - π)^(n - x) = (1 - 0.5)^(11 - 11) = 0^0 = 1
4. Plug the values into the formula:
P(x) = n! / x! (n - x)! π^x (1 - π)^(n - x)
= (39,916,800) / (39,916,800)(1)(0.00048828125)(1)
= 0.00048828125
Therefore, the probability that a stock will show a decrease in its closing price on eleven consecutive days is approximately 0.0005 (rounded to four decimal places).
To calculate the probability of a stock showing a decrease in its closing price on eleven consecutive days, you can use the binomial probability distribution formula:
P(x) = nCx * p^x * (1 - p)^(n - x)
Where:
- n is the number of trials (days in this case)
- x is the number of successes (decreases in this case)
- p is the probability of success on a single trial (probability of a decrease in this case)
In this case:
- n = 11
- x = 11 (since we want to calculate the probability of 11 consecutive decreases)
- p = 0.5 (since both increase and decrease are assumed to be equally likely, so the probability of a decrease is 0.5)
Now, let's calculate the probability:
P(11) = 11C11 * 0.5^11 * (1 - 0.5)^(11 - 11)
First, let's calculate 11C11:
11C11 = 11! / (11!(11-11)!)
= 11! / (11!(0)!)
= 11! / (11! * 1)
= 1
Now, let's substitute the values into the formula:
P(11) = 1 * 0.5^11 * (1 - 0.5)^(11 - 11)
= 0.5^11 * 0.5^0
= 0.5^11
= 0.00048828125
So, the probability that a stock will show a decrease in its closing price on eleven consecutive days is approximately 0.0005, rounded to four decimal places.
Therefore, the correct answer is 0.0005.