Answer :
To find the probability that a senior takes the bus to school every day, given that he or she has a driver's license, we can use the concept of conditional probability. Here's how you can solve it step-by-step:
1. Identify the Given Probabilities:
- [tex]\( P(D) \)[/tex]: Probability that a senior has a driver's license. This is [tex]\(84\%\)[/tex] or [tex]\(0.84\)[/tex].
- [tex]\( P(B) \)[/tex]: Probability that a senior takes the bus every day. This is [tex]\(16\%\)[/tex] or [tex]\(0.16\)[/tex].
- [tex]\( P(D \cap B) \)[/tex]: Probability that a senior both has a driver's license and takes the bus every day. This is [tex]\(14\%\)[/tex] or [tex]\(0.14\)[/tex].
2. Using the Formula for Conditional Probability:
- The formula for the conditional probability of [tex]\( B \)[/tex] given [tex]\( D \)[/tex] is:
[tex]\[
P(B \mid D) = \frac{P(D \cap B)}{P(D)}
\][/tex]
- This formula tells us the probability that a student takes the bus every day, knowing that they have a driver's license.
3. Perform the Calculation:
- Substitute the values into the formula:
[tex]\[
P(B \mid D) = \frac{0.14}{0.84}
\][/tex]
4. Convert to Percentage and Round:
- Calculate the result and convert it to a percentage:
[tex]\[
P(B \mid D) = 0.1667 \text{ (approximately)}
\][/tex]
- Convert this to a percentage by multiplying by 100:
[tex]\[
0.1667 \times 100 \approx 16.67\%
\][/tex]
- Round this to the nearest whole percent:
[tex]\[
\approx 17\%
\][/tex]
Therefore, to the nearest whole percent, the probability that a senior takes the bus to school every day, given that he or she has a driver's license, is [tex]\(17\%\)[/tex].
1. Identify the Given Probabilities:
- [tex]\( P(D) \)[/tex]: Probability that a senior has a driver's license. This is [tex]\(84\%\)[/tex] or [tex]\(0.84\)[/tex].
- [tex]\( P(B) \)[/tex]: Probability that a senior takes the bus every day. This is [tex]\(16\%\)[/tex] or [tex]\(0.16\)[/tex].
- [tex]\( P(D \cap B) \)[/tex]: Probability that a senior both has a driver's license and takes the bus every day. This is [tex]\(14\%\)[/tex] or [tex]\(0.14\)[/tex].
2. Using the Formula for Conditional Probability:
- The formula for the conditional probability of [tex]\( B \)[/tex] given [tex]\( D \)[/tex] is:
[tex]\[
P(B \mid D) = \frac{P(D \cap B)}{P(D)}
\][/tex]
- This formula tells us the probability that a student takes the bus every day, knowing that they have a driver's license.
3. Perform the Calculation:
- Substitute the values into the formula:
[tex]\[
P(B \mid D) = \frac{0.14}{0.84}
\][/tex]
4. Convert to Percentage and Round:
- Calculate the result and convert it to a percentage:
[tex]\[
P(B \mid D) = 0.1667 \text{ (approximately)}
\][/tex]
- Convert this to a percentage by multiplying by 100:
[tex]\[
0.1667 \times 100 \approx 16.67\%
\][/tex]
- Round this to the nearest whole percent:
[tex]\[
\approx 17\%
\][/tex]
Therefore, to the nearest whole percent, the probability that a senior takes the bus to school every day, given that he or she has a driver's license, is [tex]\(17\%\)[/tex].
