Question

use the table below to answer the questions.
if a student is randomly selected, what is the probability they are a girl?
if a student is randomly selected, what is the probability they are a boy that can bike to school?
if a student is randomly selected, what is the probability they are a girl or can bike to school?
selected student is a boy, what is the probability they can bike to school?

Explanation:

Step1: Recall the formula for probability

The formula for probability is \(P(A)=\frac{n(A)}{n(S)}\), where \(n(A)\) is the number of elements in event \(A\) and \(n(S)\) is the number of elements in the sample space.

Step2: Identify the number of girls and total number of students

From the table, the number of girls \(n(\text{Girls}) = 19\) and the total number of students \(n(\text{Total})=30\).

Step3: Calculate the probability of selecting a girl

Using the formula \(P(\text{Girl})=\frac{19}{30}\)

Step4: Identify the number of boys who can bike and total number of students

From the table, the number of boys who can bike \(n(\text{Boys who can bike}) = 7\) and the total number of students \(n(\text{Total}) = 30\).

Step5: Calculate the probability of selecting a boy who can bike

Using the formula \(P(\text{Boy who can bike})=\frac{7}{30}\)

Step6: Use the formula for \(P(A\cup B)\)

The formula \(P(A\cup B)=P(A)+P(B)-P(A\cap B)\). Let \(A\) be the event of selecting a girl and \(B\) be the event of selecting a student who can bike. \(P(A)=\frac{19}{30}\), \(P(B)=\frac{16}{30}\), and \(P(A\cap B)=\frac{9}{30}\)

Step7: Calculate \(P(\text{Girl}\cup\text{Can bike})\)

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Step8: Use the formula for conditional probability

The formula for conditional probability \(P(B|A)=\frac{P(A\cap B)}{P(A)}\). Here, \(A\) is the event of selecting a boy and \(B\) is the event of selecting a student who can bike. \(n(\text{Boys}) = 11\) (total number of boys) and \(n(\text{Boys who can bike})=7\). So \(P(\text{Can bike}|\text{Boy})=\frac{7}{11}\) (but if we use the formula \(P(B|A)=\frac{n(A\cap B)}{n(A)}\), \(n(A\cap B) = 7\) (number of boys who can bike) and \(n(A)=11\) (number of boys))

Answer:

• Probability of selecting a girl: \(\frac{19}{30}\)
• Probability of selecting a boy who can bike: \(\frac{7}{30}\)
• Probability of selecting a girl or a student who can bike: \(\frac{26}{30}\)
• Probability that a selected boy can bike: \(\frac{7}{11}\)