Question

a group of athletic people were asked whether they played soccer or basketball. the table shows the probabilities of results. which statement is true? playing soccer and basketball are not independent since ( p(\text{soccer}|\text{basketball})
eq p(\text{soccer}) ) and ( p(\text{basketball}|\text{soccer})
eq p(\text{basketball}) )

Explanation:

Step1: Calculate \(P(\text{soccer}|\text{basketball})\)

By the formula \(P(A|B)=\frac{P(A\cap B)}{P(B)}\), here \(A\) is "soccer" and \(B\) is "basketball". \(P(\text{soccer}\cap\text{basketball}) = 0.3\), \(P(\text{basketball})=0.7\), so \(P(\text{soccer}|\text{basketball})=\frac{0.3}{0.7}=\frac{3}{7}\approx0.43\)

Step2: Compare with \(P(\text{soccer})\)

\(P(\text{soccer}) = 0.5\). Since \(\frac{3}{7}
eq0.5\)

Step3: Calculate \(P(\text{basketball}|\text{soccer})\)

By the formula \(P(B|A)=\frac{P(A\cap B)}{P(A)}\), \(P(\text{basketball}|\text{soccer})=\frac{0.3}{0.5}=0.6\)

Step4: Compare with \(P(\text{basketball})\)

\(P(\text{basketball}) = 0.7\). Since \(0.6
eq0.7\)

Answer:

Playing soccer and basketball are not independent since \(P(\text{soccer}|\text{basketball})
eq P(\text{soccer})\) and \(P(\text{basketball}|\text{soccer})
eq P(\text{basketball})\)