Let's solve this step by step.
### Step (a)
We need to find the probability of a student spending the money given that the student was given four quarters.
To do this, we divide the number of students who spent the money by the total number of students who were given four quarters.
- Number of students who purchased gum with four quarters: 29
- Total number of students given four quarters: 29 (purchased gum) + 17 (kept the money) = 46
The probability is calculated as follows:
[tex]\[ \text{Probability (spent with quarters)} = \frac{\text{Number of students who purchased gum}}{\text{Total number of students given four quarters}} = \frac{29}{46} \][/tex]
From the given answer:
[tex]\[ \text{Probability (spent with quarters)} \approx 0.63 \][/tex]
So, the probability is [tex]\( 0.63 \)[/tex].
### Step (b)
Next, we need to find the probability of a student keeping the money given that the student was given four quarters.
To do this, we divide the number of students who kept the money by the total number of students who were given four quarters.
- Number of students who kept money with four quarters: 17
- Total number of students given four quarters: 29 (purchased gum) + 17 (kept the money) = 46
The probability is calculated as follows:
[tex]\[ \text{Probability (kept with quarters)} = \frac{\text{Number of students who kept the money}}{\text{Total number of students given four quarters}} = \frac{17}{46} \][/tex]
From the given answer:
[tex]\[ \text{Probability (kept with quarters)} \approx 0.37 \][/tex]
So, the probability is [tex]\( 0.37 \)[/tex].
### Step (c)
Finally, we need to decide what the preceding results suggest:
- A. A student given four quarters is more likely to have kept the money.
- B. A student given four quarters is more likely to have spent the money than a student given a [tex]$\$[/tex] 1[tex]$ bill.
- C. A student given four quarters is more likely to have spent the money.
- D. A student given four quarters is more likely to have kept the money than a student given a $[/tex]\[tex]$ 1$[/tex] bill.
From the results:
- Probability of spending the money with four quarters: 0.63
- Probability of keeping the money with four quarters: 0.37
Since the probability of spending the money with four quarters (0.63) is higher than the probability of keeping it (0.37), we can conclude:
B. A student given four quarters is more likely to have spent the money than a student given a [tex]$\$[/tex] 1$ bill.
Therefore, the answers are:
- Step (a): The probability is [tex]\(0.63\)[/tex].
- Step (b): The probability is [tex]\(0.37\)[/tex].
- Step (c): The correct option is B.
