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

Answer:

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

Step-by-step 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 :

$P(RedColorMorph)=\frac{47}{100}=0.47$

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 $nCx=\frac{n!}{x!(n-x)!}$    ⇒

$P(X=x)=(30Cx).(0.47)^{x}.(0.53)^{30-x}$

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

$P(X\leq 10)=0.093031581$