Answered

Roxanne likes to fish. She estimates that 30% of the fish she catches are trout, 20% are bass, and 10% are perch. She designs a simulation.

- Let 0, 1, and 2 represent trout.
- Let 3 and 4 represent bass.
- Let 5 represent perch.
- Let 6, 7, 8, and 9 represent other fish.

The table shows the simulation results.

\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}

What is the estimated probability that at least one of the next four fish Roxanne catches will be a perch?

A. 65%
B. 35%
C. 70%
D. 60%



Answer :

To answer the question about the estimated probability that at least one of the next four fish Roxanne catches will be a perch, we need to follow several steps.

Step 1: Analyze the simulation results to determine the frequency of each type of fish.

From the table, we can extract a list of all fish numbers:

```
7888, 2635, 2961, 2053, 2095,
4526, 6994, 4348, 3087, 7282,
8323, 3579, 3840, 6839, 5168,
0585, 1780, 3363, 7683, 2921
```

Step 2: Categorize the digits as per Roxanne's encoding:

- Trout: 0, 1, 2 (30%)
- Bass: 3, 4 (20%)
- Perch: 5 (10%)
- Other Fish: 6, 7, 8, 9 (40%)

Step 3: Count each type of fish from the digits in the table:

- Trout: 20
- Bass: 18
- Perch: 8
- Other Fish: 34

Step 4: Calculate the total number of fish:

Total number of fish = 80

Step 5: Determine the proportions of each type of fish:

- Proportion of Trout: [tex]\( \frac{20}{80} = 0.25 \)[/tex] (25%)
- Proportion of Bass: [tex]\( \frac{18}{80} = 0.225 \)[/tex] (22.5%)
- Proportion of Perch: [tex]\( \frac{8}{80} = 0.1 \)[/tex] (10%)
- Proportion of Other Fish: [tex]\( \frac{34}{80} = 0.425 \)[/tex] (42.5%)

Step 6: Calculate the probability that at least one of the next four fish Roxanne catches will be a perch.

- The probability of not catching a perch in a single catch is [tex]\( 1 - 0.1 = 0.9 \)[/tex].
- The probability of not catching a perch in each of the next four catches is [tex]\( (0.9)^4 = 0.6561 \)[/tex].
- Therefore, the probability of catching at least one perch in the next four catches is:
[tex]\[ 1 - 0.6561 = 0.3439 \approx 34.39% \][/tex]

So, the estimated probability that at least one of the next four fish Roxanne catches will be a perch is approximately 34.39%. Therefore, the closest answer is:

B. [tex]\(35 \%\)[/tex]