Question

A population of red-bellied snakes is known to have a ratio of grey color morph to red color morph of 53:47. You wish to know the probability of selecting a random sample of 30 snakes containing 10 or fewer red morph individuals. Use a statistical program to determine the exact probability of getting 10 or fewer grey morph individuals.

Answer 1

The probability is
`P(X\leq 10)=0.093031581`

Explanation:

We know that the ratio of grey color morph to red color morph is
`53:47`

This can be written in terms of probability as :

`P(RedColorMorph)=(47)/(100)=0.47`

This means that the probability of obtain a red color morph snake in a random sample of 100 snakes is
`0.47` (If I only randomly select one snake).

Now, the number
`X` of red color morph snakes in a random sample ''n'' can be modeled as a binomial random variable. Where ''p'' is the success probability (In our case, the probability from obtain one red color morph snake out of 100 snakes) ⇒

`X:` ''Number of red color morph snakes in the sample''

`p=0.47`

In our case,
`n=30`

⇒
`X` ~ Bi (n,p) ⇒
`X` ~ Bi (30,0.47)

The probability function for
`X` is :

`P(X=x)=(nCx).p^(x).(1-p)^(n-x)`

Where
`nCx` is the combinatorial number define as
`nCx=(n!)/(x!(n-x)!)` ⇒

`P(X=x)=(30Cx).(0.47)^(x).(0.53)^(30-x)`

We need to calculate the probability of
`P(X\leq 10)`

This probability is equal to :

`P(X\leq 10)=P(X=0)+P(X=1)+P(X=2)+P(X=3)+P(X=4)+P(X=5)+P(X=6)+P(X=7)+P(X=8)+P(X=9)+P(X=10)`

For example,

`P(X=6)=(30C6).(0.47)^(6).(0.53)^(24)=0.0015446`

We need to calculate all the terms of the sum and then calculate
`P(X\leq 10)`

If we use any statistical program we will find that

`P(X\leq 10)=0.093031581`