Scientists randomly caught 30 fruit bats at an orchard. They tagged the fruit
bats and then released them. Several weeks later, they captured 20 fruit bats at the same location. They found that 4 of those fruit bats
had tags. Assume the population of fruit bats does not change. About how many fruit bats
are at that orchard?
There are two ways to solve this problem.
A.
The intuitive (and fast) approximation is by prorating the ratio, assuming that the ratio is constant.
Let N=total population, then
N/30=20/4 => N=150
B.
The more accurate (and elaborate) way is to use the hypergeometric distribution. This distribution is typically used in instances where sampling is done without replacement, such as drawing cards out of a deck.
Let
A=recapture size = 20
B=uncaptured size = population -20
m=number of marked in sample (4)
n=number of marked in the wild (30-4=26)
then
P(X=m)=C(A,m)*C(B,n)/C(A+B,m+n)
C(n,r)=combination n choose r = n!/(r!(n-r)!)
We need to find A+B.
Substituting formula,
P(x=4)=C(20,4)*C(B,26)/C(B+20,30)
The value of P(x=4) is not directly known, but the maximum likelihood is when it is at the maximum for the various values of B.
We will find that P(x=4) is at a maximum when B=129.5, which makes the total population A+B=20+129.5=149.5, quite close to our initial estimate.
Well, it seems that the scientists have found a way to tag fruit bats and make them want to return to the orchard for further study. Those bats are quite dedicated to science! Now, if we use our mathematical skills, we can estimate the total number of fruit bats at the orchard.
So, let's assume that the ratio of tagged bats to the captured bats is the same as the ratio of tagged bats to the total population. If 4 out of the 20 captured bats have tags, then we can set up the proportion:
4 tagged bats / 20 captured bats = x tagged bats / total population
Now, let's solve for x, the number of tagged bats in the total population. Cross-multiplying gives us:
20x = 4 * total population
Dividing both sides by 20, we get:
x = (4 * total population) / 20
Since we know that there were initially 30 tagged bats, we can set up another equation:
x = 30 tagged bats / total population
Now, we can equate these two expressions for x:
(4 * total population) / 20 = 30 / total population
Cross-multiplying again:
4 * total population * total population = 20 * 30
Dividing both sides by 4:
total population * total population = 150
Taking the square root of both sides:
total population ≈ √(150) ≈ 12.25
Now, since we can't have a fraction of a bat, we'll round this down to the nearest whole number. Therefore, there are approximately 12 fruit bats at that orchard.
Please bear in mind that this is just an estimate using the information provided. The actual number of fruit bats could be different, but I hope this gives you a rough idea!
To estimate the number of fruit bats at the orchard, we can use a proportion based on the ratio of tagged bats to the total number of bats captured in the second sample.
Let's represent the total number of fruit bats in the orchard as "x". We know that:
- In the first sample, they caught 30 bats and tagged all of them.
- In the second sample, they caught 20 bats and 4 of them had tags.
So, we can set up the following proportion:
(tags in second sample) / (total bats in second sample) = (total tagged bats) / (total bats in orchard)
4 / 20 = 30 / x
Cross-multiplying:
4x = 20 * 30
4x = 600
Dividing both sides by 4:
x = 600 / 4
x = 150
Therefore, there are approximately 150 fruit bats at the orchard.
To estimate the number of fruit bats at the orchard, we can use the concept of mark and recapture.
In this scenario, the scientists initially captured and tagged 30 fruit bats. Later, they captured another 20 bats and found that 4 of them had tags.
To estimate the total population, we can set up a proportion using the mark and recapture formula:
(Number of tagged bats in the first sample) / (Total population) = (Number of tagged bats in the second sample) / (Number of bats in the second sample)
Plugging in the values we have:
30 / Total population = 4 / 20
Now, we can cross multiply and solve for the total population:
4(Total population) = 30 * 20
Total population = (30 * 20) / 4
Total population ≈ 150 fruit bats
So, there are approximately 150 fruit bats at that orchard.