word problem for Advanced Mathematicians only

Question

If you must make a decision that will have a 50% chance of working in your favor as a standalone, and there is also an added 50% chance that surrounding events will help your decision work in your favor, what is the probability of your decision working in your favor throughout 24 decisions? Please state the answer in percentage form as well as a fraction.

Answer 1

 

The surrounding events having nothing in common with the decision working out in  favorable outcome outright on it's own accord and merit as a standalone. They are independent events.

 

The probability is :   

 

  prob( favorable outcome outright) + prob( favorable outcome AND surrounding event) =

        1/2 + 1/2*1/2 =

       1/2 + 1/4 =

    3/4

 

p=3/4 = 0.75

 

Binomial p=3/4=0.75,   q = 1 - p = 0.25 = 1/4 ,  N = 24 and k=24

 

 

(N choose k) = (24 choose 24) = 1 as is q^0 = 1

 

p^24 = (3/4)^24 = 282429536481 / 281474976710656 or just over 0.1% 

and this is a BEST CASE scenario

 

It is very unlikely you will run the table.

Think of modeling the problem this way...

 

You generate two random numbers between 1 and 100.

a result of 50 or more in the first number represents the

decision having a favorable outcome outright without any external

help. A result of 50 or more of the second number represents

a favorable outcome due to the surrounding external circumstances.

 

If BOTH numbers are LESS than 50, then the decision is a FAILURE.

Generating 24 such pairs of numbers, ALL 24 ordered pairs must

have at least one coordinate greater than or equal to 50.

 

In fact, the expected value is

1/2*1/2 = 1/4 as a failure which is consistent with the 

probability calculated above. 1/4*24 = 6 such failures.

 

I simulated this experiment in excel 100 times

and there were NO sets of 24 with perfect scores.