Question

the results are shown below:

consider the following events:
a. the order was takeout.
b. the order was dine - in.
c. the order was placed on a weekday.
d. the order was placed on a weekend.
which two events are independent?
a and b
c and d
a and c
none of the above

Explanation:

Step1: Recall the formula for independent events

Two events \(E\) and \(F\) are independent if \(P(E\cap F)=P(E)\times P(F)\)

Step2: Calculate probabilities for each pair

• For events \(A\) and \(B\):

\(P(A)=\frac{100}{200} = 0.5\), \(P(B)=\frac{100}{200}=0.5\), \(P(A\cap B) = 0\) (since an order can't be both take - out and dine - in). And \(P(A)\times P(B)=0.5\times0.5 = 0.25
eq0\)

• For events \(C\) and \(D\):

\(P(C)=\frac{120}{200}=0.6\), \(P(D)=\frac{80}{200} = 0.4\), \(P(C\cap D)=0\) (an order can't be on a weekday and weekend simultaneously). And \(P(C)\times P(D)=0.6\times0.4=0.24
eq0\)

• For events \(A\) and \(C\):

\(P(A)=\frac{100}{200}=0.5\), \(P(C)=\frac{120}{200}=0.6\), \(P(A\cap C)=\frac{66}{200}=0.33\). And \(P(A)\times P(C)=0.5\times0.6 = 0.3
eq0.33\)

Answer:

None of the above