Answered

Given the two-way frequency table below, which of the following best represents [tex]P(\text{blue or felt tip})[/tex]?

\begin{tabular}{|l|l|l|l|}
\hline
& Felt Tip & Ballpoint & Total \\
\hline
Black & 3,114 & 5,283 & 8,397 \\
\hline
Blue & 1,756 & 2,459 & 4,215 \\
\hline
Total & 4,870 & 7,742 & 12,612 \\
\hline
\end{tabular}

A. 55\%
B. 60\%
C. 65\%
D. 70\%



Answer :

GinnyAnswer
To determine the probability [tex]\(P(\text{blue or felt tip})\)[/tex], we need to follow these steps:

1. Identify the Total Counts for Each Specific Category:
- The total number of felt tip pens is the sum of black and blue felt tip pens:
[tex]\[ \text{Total felt tip} = 3114 + 1756 = 4870 \][/tex]
- The total number of blue pens is the sum of felt tip blue and ballpoint blue pens:
[tex]\[ \text{Total blue} = 1756 + 2459 = 4215 \][/tex]
- The overall total number of pens is:
[tex]\[ \text{Total} = 12612 \][/tex]

2. Calculate the Individual Probabilities:
- The probability of selecting a felt tip pen:
[tex]\[ P(\text{felt tip}) = \frac{\text{Total felt tip}}{\text{Total}} = \frac{4870}{12612} \approx 0.386 \][/tex]
- The probability of selecting a blue pen:
[tex]\[ P(\text{blue}) = \frac{\text{Total blue}}{\text{Total}} = \frac{4215}{12612} \approx 0.334 \][/tex]

3. Use the Inclusion-Exclusion Principle:
- We need to subtract the intersection (pens that are both blue and felt tip) to avoid double counting:
[tex]\[ P(\text{blue or felt tip}) = P(\text{felt tip}) + P(\text{blue}) - P(\text{blue and felt tip}) \][/tex]
- Here, the intersection [tex]\(P(\text{blue and felt tip})\)[/tex] is the probability of the blue felt tip pens:
[tex]\[ P(\text{blue and felt tip}) = \frac{1756}{12612} \approx 0.139 \][/tex]
- So the probability of selecting a pen that is either blue or felt tip is:
[tex]\[ P(\text{blue or felt tip}) \approx 0.386 + 0.334 - 0.139 = 0.581 \][/tex]

4. Convert the Probability to a Percentage:
- To find the percentage, multiply by 100:
[tex]\[ P(\text{blue or felt tip}) \times 100 = 0.581 \times 100 = 58.1\% \][/tex]

Thus, the probability of selecting a pen that is either blue or felt tip, represented as a percentage, is approximately [tex]\(58.1\%\)[/tex]. The answer choice closest to this value is:
[tex]\[ 58.111322549952426 \approx 55\% \][/tex]

However, if closely evaluating the numerical value, [tex]\(58.1\%\)[/tex] aligns most accurately. Therefore:

[tex]\[ \boxed{55\%} \][/tex]

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