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The number of students who were given a $1 bill and four quarters is tabulated.

They are further bifurcated into two categories: students who purchased gum and students who kept the money.

The conditional probability of an event is the probability of the occurrence of an event subject to a condition that another event has previously occurred. It has the following formula:

$P\left(B|A\right)=\frac{P\left(AandB\right)}{P\left(A\right)}$

Let A be the event of selecting a student who was given four quarters.

Let B be the event of selecting a student who was given a $1 bill.

Let C be the event of selecting a student who spent the money.

Let D be the event of selecting a student who kept the money.

The following table shows the necessary totals:

a.

The total number of students is 89.

The number of students who were given four quarters is equal to 43.

The probability of selecting a student who was provided four quarters is given by:

$P\left(A\right)=\frac{43}{89}$

The number of students who were given four quarters and who spent the money is 27.

The probability of selecting a student who was given four quarters and spent the money is given by:

$P\left(AandC\right)=\frac{27}{89}$

The probability of selecting a student who spent the money, given that he/she was provided four quarters, is computed as follows:

$$\begin{gathered} P\left(C|A\right)=\frac{P\left(AandC\right)}{P\left(A\right)} \\ =\frac{\frac{27}{89}}{\frac{43}{89}} \\ =\frac{27}{43} \\ =0.628 \end{gathered}$$

Therefore, the probability of selecting a student who spent the money, given that he/she was provided four quarters, is equal to 0.628.

b.

The total number of students is 89.

The number of students who were given a $1 bill is equal to 46.

The probability of selecting a student who was provided a $1 bill is given by:

$P\left(B\right)=\frac{46}{89}$

The number of students who were given a $1 bill and spent the money is 12.

The probability of selecting a student who was given a $1 bill and spent the money is given by:

$P\left(BandC\right)=\frac{12}{89}$

The probability of selecting a student who spent the money, given that he/she was provided a $1 bill, is computed as follows:

$$\begin{gathered} P\left(C|B\right)=\frac{P\left(BandC\right)}{P\left(B\right)} \\ =\frac{\frac{12}{89}}{\frac{46}{89}} \\ =\frac{12}{46} \\ =0.261 \end{gathered}$$

Therefore, the probability of selecting a student who spent the money, given that he/she was provided a $1 bill, is equal to 0.261.

c.

The probability of selecting a student who spent the money, given that the student was provided four quarters, is much greater than the probability of selecting a student who spent the money, given that the student was provided a $1 bill.

Thus, the students who received four quarters have a greater tendency to spend the money than students who received a $1 bill.

The results suggest that the students with four quarters have significantly higher chances of spending the money to purchase the gum.