Answer :
To determine the estimated probability that at least one of the next four fish Roxanne catches will be a perch, let's follow a detailed step-by-step solution.
### Step 1: Convert the simulation results into a list of digits
First, we need to extract all the digits from the given simulation results:
[tex]\[ \begin{tabular}{|l|l|l|l|l|} \hline \multicolumn{5}{|c|}{Simulation Results} \\ \hline 7888 & 2635 & 2961 & 2053 & 2095 \\ \hline 4526 & 6994 & 4348 & 3087 & 7282 \\ \hline 8323 & 3579 & 3840 & 6839 & 5168 \\ \hline 0585 & 1780 & 3363 & 7683 & 2921 \\ \hline \end{tabular} \][/tex]
This gives us the following sequence of digits:
[tex]\[ 7, 8, 8, 8, 2, 6, 3, 5, 2, 9, 6, 1, 2, 0, 5, 3, 2, 0, 9, 5, 4, 5, 2, 6, 6, 9, 9, 4, 4, 3, 4, 8, 3, 0, 8, 7, 7, 2, 8, 2, 8, 3, 2, 3, 3, 5, 7, 9, 3, 8, 4, 0, 6, 8, 3, 9, 5, 1, 6, 8, 0, 5, 8, 5, 1, 7, 8, 0, 3, 3, 3, 6, 3, 7, 6, 8, 3, 2, 9, 2, 1 \][/tex]
### Step 2: Count the total number of digits
Next, we need to calculate the total number of digits:
Total number of digits = 4 rows × 5 columns × 4 digits each = 80 digits
### Step 3: Count the number of '5's
The digit '5' represents perch. We need to count the number of times '5' appears in the list. By counting manually or programmatically, we find that:
Number of '5's (perch) = 9
### Step 4: Calculate the probability of catching a perch
Probability of catching a perch, [tex]\(P(\text{perch})\)[/tex] can be calculated as:
[tex]\[ P(\text{perch}) = \frac{\text{Number of '5's}}{\text{Total number of digits}} = \frac{9}{80} = 0.1125 \][/tex]
### Step 5: Calculate the probability of NOT catching a perch
Probability of NOT catching a perch, [tex]\(P(\text{not perch})\)[/tex], is:
[tex]\[ P(\text{not perch}) = 1 - P(\text{perch}) = 1 - 0.1125 = 0.8875 \][/tex]
### Step 6: Calculate the probability that all of the next four fish are NOT perch
We need to raise the probability of NOT catching a perch to the power of 4, since we are considering four fish:
[tex]\[ P(\text{none perch}) = (P(\text{not perch}))^4 = (0.8875)^4 \approx 0.6215 \][/tex]
### Step 7: Calculate the probability that at least one of the next four fish is a perch
Finally, the probability that at least one fish in the next four is a perch is:
[tex]\[ P(\text{at least one perch}) = 1 - P(\text{none perch}) = 1 - 0.6215 = 0.3785 \approx 38\% \][/tex]
The closest percentage from the given options is:
D. [tex]\(35 \%\)[/tex]
Thus, the correct answer is:
[tex]\[ \boxed{35\%} \][/tex]
### Step 1: Convert the simulation results into a list of digits
First, we need to extract all the digits from the given simulation results:
[tex]\[ \begin{tabular}{|l|l|l|l|l|} \hline \multicolumn{5}{|c|}{Simulation Results} \\ \hline 7888 & 2635 & 2961 & 2053 & 2095 \\ \hline 4526 & 6994 & 4348 & 3087 & 7282 \\ \hline 8323 & 3579 & 3840 & 6839 & 5168 \\ \hline 0585 & 1780 & 3363 & 7683 & 2921 \\ \hline \end{tabular} \][/tex]
This gives us the following sequence of digits:
[tex]\[ 7, 8, 8, 8, 2, 6, 3, 5, 2, 9, 6, 1, 2, 0, 5, 3, 2, 0, 9, 5, 4, 5, 2, 6, 6, 9, 9, 4, 4, 3, 4, 8, 3, 0, 8, 7, 7, 2, 8, 2, 8, 3, 2, 3, 3, 5, 7, 9, 3, 8, 4, 0, 6, 8, 3, 9, 5, 1, 6, 8, 0, 5, 8, 5, 1, 7, 8, 0, 3, 3, 3, 6, 3, 7, 6, 8, 3, 2, 9, 2, 1 \][/tex]
### Step 2: Count the total number of digits
Next, we need to calculate the total number of digits:
Total number of digits = 4 rows × 5 columns × 4 digits each = 80 digits
### Step 3: Count the number of '5's
The digit '5' represents perch. We need to count the number of times '5' appears in the list. By counting manually or programmatically, we find that:
Number of '5's (perch) = 9
### Step 4: Calculate the probability of catching a perch
Probability of catching a perch, [tex]\(P(\text{perch})\)[/tex] can be calculated as:
[tex]\[ P(\text{perch}) = \frac{\text{Number of '5's}}{\text{Total number of digits}} = \frac{9}{80} = 0.1125 \][/tex]
### Step 5: Calculate the probability of NOT catching a perch
Probability of NOT catching a perch, [tex]\(P(\text{not perch})\)[/tex], is:
[tex]\[ P(\text{not perch}) = 1 - P(\text{perch}) = 1 - 0.1125 = 0.8875 \][/tex]
### Step 6: Calculate the probability that all of the next four fish are NOT perch
We need to raise the probability of NOT catching a perch to the power of 4, since we are considering four fish:
[tex]\[ P(\text{none perch}) = (P(\text{not perch}))^4 = (0.8875)^4 \approx 0.6215 \][/tex]
### Step 7: Calculate the probability that at least one of the next four fish is a perch
Finally, the probability that at least one fish in the next four is a perch is:
[tex]\[ P(\text{at least one perch}) = 1 - P(\text{none perch}) = 1 - 0.6215 = 0.3785 \approx 38\% \][/tex]
The closest percentage from the given options is:
D. [tex]\(35 \%\)[/tex]
Thus, the correct answer is:
[tex]\[ \boxed{35\%} \][/tex]
