Probability - bionomial distribution homework?
In a certain population, 15% of the people have Rh-negative blood. A blood bank serving this population receives 100 blood donors on a particular day.
a) Let X be the number of donors in the sample with Rh-negative blood. What is the exact distribution of X?
So I get that this is a binomial distribution. But I don't understand what the question means when it asks what is the exact distribution of X? How would I find the answer to a)?
- AlanLv 76 months agoFavorite Answer
It does seem like too much unless you are allowed to use a spreadsheet, but then
of course then the answer would get rounded making then inexact.
Exact normally means no rounding to keep all answers as fraction.
so if they will just accept a formula which is good for k = 0 to 100
then, it is not as hard as we think.
so one exact would be
so using * to mean multiply
So this is an exact formula for all 100 values
P(x = k) = (100 k) *(15^k)*(85^(100-k)/100^100 for k = 0 to 100
elsewhere P(x = k) = 0
P(x= k) = (100! / (k! *(100-k)! )*(15^k)*(85^(100-k)/100^100 for k = 0 to 100
elsewhere P(x=k) = 0
X would go from 0 to 100
P(x=0) = ( 100 0 )*(15/100)^0 (85/100)^100 = (85^100) / ( 100)^100
P(x = 1) = (100 1) *(15/100)^1*(85/100)^99 = 10015*85^99/100^100
P(x = 2) = (100 2)*(15/100)^2 (85/100)^(98) = 50*99*(85^2)*(15^98)/ 100^100
P(x = k) = (100 k) *(15^k)*(85^(100-k)/100^100
P(x = k) = ( 100! / (k! *(100-k)! ) )* *(15^k)*(85^(100-k)/100^100
P(x = 98) = 50*99*(85^98)*(15^2)/ (100^100)
P(x = 99) = 100*(85^99)*(15)/ (100^100)
P(x = 100) = (85^100)/ (100^100)
since some of these number are extremely large, I cannot see you multiply them out.
so you would have to leave things as fractions with exponents
Also, I cannot see you determining all of the factorial due to their sizes.
so either you would have to leave as a combinations (100 5) or 100!/ (5! *95!)
so exact usually means no rounding so you would have to leave as factorials or combinations
and fractional exponents
Also, there is the question of how much simplifying you would want to do.
- davidLv 76 months ago
It appears that the problem wants you to find the probablitiy of every value of X .. X = 0 would mean that there are NO doners with Rh- blood /// then X = 1 would mean that exactly 1 donor was Rh-
and X = 2 ... exactly 2 donors have Rh- and on and on until finally x = 100 means that all 100 donors have Rh- blood.
=== seems like a lot of useless work just to emphasize to you the way to calculate binomial probabilities. === Good luck