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A bag contains 10 marbles: 3 red, 5 blue, and 2 violet. One marble is drawn and then replaced, and this action is repeated 10 times. The results are recorded in the table.

\begin{tabular}{|c|c|c|}
\hline
Event & Frequency & Probability \\
\hline
Blue & 5 & [tex]$\frac{5}{10}$[/tex] \\
\hline
Red & 1 & [tex]$\frac{3}{10}$[/tex] \\
\hline
Violet & 4 & [tex]$\frac{2}{10}$[/tex] \\
\hline
\end{tabular}

Which of the given statements is true?
A) The experimental probability of drawing blue is [tex]$\frac{1}{2}$[/tex].
B) The experimental probability of drawing red is 1.
C) The theoretical probability of drawing red is [tex]$\frac{3}{10}$[/tex].
D) The theoretical probability of drawing violet is 0.



Answer :

GinnyAnswer
Let's analyze the given scenario and determine the truth values of the statements provided:

1. Total number of marbles in the bag:
- Red marbles: 3
- Blue marbles: 5
- Violet marbles: 2
- Total marbles: \(3 + 5 + 2 = 10\)

2. Experimental probabilities:

- The frequency of choosing a blue marble is 5 out of 10 draws.
- Experimental probability of drawing blue:
[tex]\[ \text{Probability} = \frac{5}{5 + 1 + 4} = \frac{5}{10} = 0.5 \][/tex]

- The frequency of choosing a red marble is 1 out of 10 draws.
- Experimental probability of drawing red:
[tex]\[ \text{Probability} = \frac{1}{5 + 1 + 4} = \frac{1}{10} = 0.1 \][/tex]

- The frequency of choosing a violet marble is 4 out of 10 draws.
- Experimental probability of drawing violet:
[tex]\[ \text{Probability} = \frac{4}{5 + 1 + 4} = \frac{4}{10} = 0.4 \][/tex]

3. Theoretical probabilities:

- The probability of drawing a blue marble:
[tex]\[ \text{Probability} = \frac{\text{Number of blue marbles}}{\text{Total number of marbles}} = \frac{5}{10} = 0.5 \][/tex]

- The probability of drawing a red marble:
[tex]\[ \text{Probability} = \frac{\text{Number of red marbles}}{\text{Total number of marbles}} = \frac{3}{10} = 0.3 \][/tex]

- The probability of drawing a violet marble:
[tex]\[ \text{Probability} = \frac{\text{Number of violet marbles}}{\text{Total number of marbles}} = \frac{2}{10} = 0.2 \][/tex]

Now, let's evaluate the statements:

Statement A: The experimental probability of drawing blue is \( \frac{1}{2} \).

- From the experimental probability calculation, we found it to be 0.5. This is equivalent to \( \frac{1}{2} \).
- True

Statement B: The experimental probability of drawing red is 1.

- The experimental probability of drawing red was calculated to be 0.1.
- False

Statement C: The theoretical probability of drawing red is \( \frac{1}{10} \).

- The theoretical probability of drawing red was calculated to be 0.3.
- False

Statement D: The theoretical probability of drawing violet is 0.

- The theoretical probability of drawing violet was calculated to be 0.2.
- False

Based on the analysis, the true statement is A.

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