Question

joy owns an art gallery that only displays paintings and photographs. some of the pieces on display depict people. the rest depict landscapes. joy chooses a piece at random from among all those on display. let a be the event that the chosen piece is a photograph. let b be the event that the chosen piece depicts a landscape. are events a and b independent? complete the sentence. events a and b must independent because the probability of a

Explanation:

Step1: Calculate \(P(A)\)

The total number of pieces is \(n = 90\). The number of photographs is \(n(A)=50\). So \(P(A)=\frac{50}{90}=\frac{5}{9}\)

Step2: Calculate \(P(B)\)

The number of landscape - themed pieces is \(n(B) = 35\). So \(P(B)=\frac{35}{90}=\frac{7}{18}\)

Step3: Calculate \(P(A\cap B)\)

The number of landscape - themed photographs is \(n(A\cap B)=17\). So \(P(A\cap B)=\frac{17}{90}\)

Step4: Check the independence condition \(P(A\cap B)=P(A)\times P(B)\)

\(P(A)\times P(B)=\frac{5}{9}\times\frac{7}{18}=\frac{35}{162}\approx0.216\) and \(P(A\cap B)=\frac{17}{90}\approx0.189\)

Since \(P(A\cap B)
eq P(A)\times P(B)\)

Answer:

Events \(A\) and \(B\) must not be independent because the probability of \(A\) and \(B\) occurring together is not equal to the product of their individual probabilities