To solve this problem, we will compute the probability of seeing at least one robin in the next five observations, based on the simulation data provided.
### Step 1: Convert the table to a list and flatten
First, let’s list all the values from the table in a single flattened list:
[tex]\[
\begin{align*}
& \text{List from Table} = [05716, 16803, 96568, 32177, 33855, 76635, 92290, 88864, 72794, 14333, 79019, 05943, 77510, 74051, 87238, 97895, 86481, 94036, 12749, 24005]
\end{align*}
\][/tex]
### Step 2: Count the total number of trials
The total number of trials can be found by counting the elements:
[tex]\[
N = 20
\][/tex]
### Step 3: Determine the robin indicator threshold
For this problem, we assume that small values (typically those that might represent some condition) are indicative of robins. Let's assume this threshold is [tex]\(10,000\)[/tex]. So any value less than 10,000 would be considered a robin's sighting.
### Step 4: Count the occurrences of robins
Count how many numbers in our list are less than [tex]\(10,000\)[/tex]:
[tex]\[
\text{Robin count} = [05716, 05943]
\][/tex]
So, the number of such values is:
[tex]\[
R = 2
\][/tex]
### Step 5: Calculate the probability of encountering a robin in any given observation
To find the probability ([tex]\(P\)[/tex]) that a single observation is a robin, we divide the number of robins by the total number of observations:
[tex]\[
P(\text{seeing a robin}) = \frac{R}{N} = \frac{2}{20} = 0.1
\][/tex]
### Step 6: Calculate the probability of seeing at least one robin in the next five trials
To determine the probability of seeing at least one robin in the next five observations, we can use the complement rule. First, calculate the probability of not seeing a robin in a single observation:
[tex]\[
P(\text{not seeing a robin}) = 1 - P(\text{seeing a robin}) = 1 - 0.1 = 0.9
\][/tex]
The probability of not seeing a robin in five consecutive observations is:
[tex]\[
P(\text{not seeing a robin in 5 trials}) = (0.9)^5
\][/tex]
Calculate this value:
[tex]\[
(0.9)^5 \approx 0.59049
\][/tex]
Finally, the probability of seeing at least one robin in five trials is the complement of the above probability:
[tex]\[
P(\text{at least one robin in 5 trials}) = 1 - (0.9)^5 = 1 - 0.59049 = 0.40951
\][/tex]
### Conclusion
Thus, the probability that at least one of the next five birds observed will be a robin is approximately [tex]\(0.40951\)[/tex], or about [tex]\(41\%\)[/tex].
