Answer :
We start by interpreting the information given about the marbles:
1. There are 20 red marbles that have a blue dot.
2. There are 15 red marbles that do not have a blue dot.
3. There are 30 white marbles and we assume that none of these white marbles have a blue dot.
With this information, we can organize the marbles into a two‐way table with the rows representing the colors (red and white) and the columns representing whether a marble has a blue dot or not.
---
Step 1. Set Up the Table
- For red marbles:
- With blue dot: 20
- Without blue dot: 15
- Total red marbles:
[tex]$$20 + 15 = 35.$$[/tex]
- For white marbles:
- With blue dot: 0
- Without blue dot: 30
- Total white marbles:
[tex]$$0 + 30 = 30.$$[/tex]
- Overall totals:
- Total marbles in the bag:
[tex]$$35 + 30 = 65.$$[/tex]
- Total marbles with a blue dot:
[tex]$$20 + 0 = 20.$$[/tex]
- Total marbles without a blue dot:
[tex]$$15 + 30 = 45.$$[/tex]
The table now looks like this:
[tex]$$
\begin{array}{ccc|c}
& \text{Blue Dot} & \text{No Blue Dot} & \text{Total} \\ \hline
\text{Red} & 20 & 15 & 35 \\
\text{White} & 0 & 30 & 30 \\ \hline
\text{Total} & 20 & 45 & 65 \\
\end{array}
$$[/tex]
---
Step 2. Calculate the Probabilities
a) The probability that a marble is red is given by
[tex]$$
P(\text{red}) = \frac{\text{Total red marbles}}{\text{Total marbles}} = \frac{35}{65} \approx 0.5385.
$$[/tex]
b) The probability that a marble is white is given by
[tex]$$
P(\text{white}) = \frac{\text{Total white marbles}}{\text{Total marbles}} = \frac{30}{65} \approx 0.4615.
$$[/tex]
c) The probability that a marble has a blue dot is
[tex]$$
P(\text{blue dot}) = \frac{\text{Marbles with blue dot}}{\text{Total marbles}} = \frac{20}{65} \approx 0.3077.
$$[/tex]
d) The probability that a marble does not have a blue dot is
[tex]$$
P(\text{no blue dot}) = \frac{\text{Marbles without blue dot}}{\text{Total marbles}} = \frac{45}{65} \approx 0.6923.
$$[/tex]
e) The probability that a marble is white given that it does not have a blue dot is a conditional probability calculated by
[tex]$$
P(\text{white} \mid \text{no blue dot}) = \frac{\text{White marbles without blue dot}}{\text{Total marbles without blue dot}} = \frac{30}{45} \approx 0.6667.
$$[/tex]
---
Final Answer Summary
- [tex]$$P(\text{red}) = \frac{35}{65} \approx 0.5385,$$[/tex]
- [tex]$$P(\text{white}) = \frac{30}{65} \approx 0.4615,$$[/tex]
- [tex]$$P(\text{blue dot}) = \frac{20}{65} \approx 0.3077,$$[/tex]
- [tex]$$P(\text{no blue dot}) = \frac{45}{65} \approx 0.6923,$$[/tex]
- [tex]$$P(\text{white} \mid \text{no blue dot}) = \frac{30}{45} \approx 0.6667.$$[/tex]
This completes the detailed step-by-step solution.
1. There are 20 red marbles that have a blue dot.
2. There are 15 red marbles that do not have a blue dot.
3. There are 30 white marbles and we assume that none of these white marbles have a blue dot.
With this information, we can organize the marbles into a two‐way table with the rows representing the colors (red and white) and the columns representing whether a marble has a blue dot or not.
---
Step 1. Set Up the Table
- For red marbles:
- With blue dot: 20
- Without blue dot: 15
- Total red marbles:
[tex]$$20 + 15 = 35.$$[/tex]
- For white marbles:
- With blue dot: 0
- Without blue dot: 30
- Total white marbles:
[tex]$$0 + 30 = 30.$$[/tex]
- Overall totals:
- Total marbles in the bag:
[tex]$$35 + 30 = 65.$$[/tex]
- Total marbles with a blue dot:
[tex]$$20 + 0 = 20.$$[/tex]
- Total marbles without a blue dot:
[tex]$$15 + 30 = 45.$$[/tex]
The table now looks like this:
[tex]$$
\begin{array}{ccc|c}
& \text{Blue Dot} & \text{No Blue Dot} & \text{Total} \\ \hline
\text{Red} & 20 & 15 & 35 \\
\text{White} & 0 & 30 & 30 \\ \hline
\text{Total} & 20 & 45 & 65 \\
\end{array}
$$[/tex]
---
Step 2. Calculate the Probabilities
a) The probability that a marble is red is given by
[tex]$$
P(\text{red}) = \frac{\text{Total red marbles}}{\text{Total marbles}} = \frac{35}{65} \approx 0.5385.
$$[/tex]
b) The probability that a marble is white is given by
[tex]$$
P(\text{white}) = \frac{\text{Total white marbles}}{\text{Total marbles}} = \frac{30}{65} \approx 0.4615.
$$[/tex]
c) The probability that a marble has a blue dot is
[tex]$$
P(\text{blue dot}) = \frac{\text{Marbles with blue dot}}{\text{Total marbles}} = \frac{20}{65} \approx 0.3077.
$$[/tex]
d) The probability that a marble does not have a blue dot is
[tex]$$
P(\text{no blue dot}) = \frac{\text{Marbles without blue dot}}{\text{Total marbles}} = \frac{45}{65} \approx 0.6923.
$$[/tex]
e) The probability that a marble is white given that it does not have a blue dot is a conditional probability calculated by
[tex]$$
P(\text{white} \mid \text{no blue dot}) = \frac{\text{White marbles without blue dot}}{\text{Total marbles without blue dot}} = \frac{30}{45} \approx 0.6667.
$$[/tex]
---
Final Answer Summary
- [tex]$$P(\text{red}) = \frac{35}{65} \approx 0.5385,$$[/tex]
- [tex]$$P(\text{white}) = \frac{30}{65} \approx 0.4615,$$[/tex]
- [tex]$$P(\text{blue dot}) = \frac{20}{65} \approx 0.3077,$$[/tex]
- [tex]$$P(\text{no blue dot}) = \frac{45}{65} \approx 0.6923,$$[/tex]
- [tex]$$P(\text{white} \mid \text{no blue dot}) = \frac{30}{45} \approx 0.6667.$$[/tex]
This completes the detailed step-by-step solution.
