Question

(c) experiment 3: a deck contains 8 cards labeled 1 through 8. a card is randomly selected and returned to the deck. the deck is shuffled. then another random selection is made.
event a: the first selection is a 5.
event b: the second selection is a 4.
determine whether events a and b are independent or dependent.
o independent o dependent
(d) experiment 4: a family has two children.
event a: the older child is a girl.
event b: both children are girls.
determine whether events a and b are independent or dependent.
o independent dependent
(e) experiment 5: a number cube with sides labeled 1 through 6 is rolled twice.
event a: the first roll is a 5.
event b: the second roll is a 2.
determine whether events a and b are independent or dependent.
o independent o dependent

Explanation:

Step1: Recall the definition of independent events

Two events \(A\) and \(B\) are independent if \(P(A\cap B)=P(A)\times P(B)\).

Step2: Analyze Experiment 3

• For Event \(A\) (first - selection is a \(5\)): \(P(A)=\frac{1}{8}\)
• For Event \(B\) (second - selection is a \(4\)): \(P(B)=\frac{1}{8}\)
• Since the card is returned to the deck and shuffled, \(P(A\cap B)=\frac{1}{8}\times\frac{1}{8}=\frac{1}{64}\)
• \(P(A)\times P(B)=\frac{1}{8}\times\frac{1}{8}=\frac{1}{64}\)
• So \(P(A\cap B) = P(A)\times P(B)\), events \(A\) and \(B\) are independent.

Step3: Analyze Experiment 4

• Let \(G\) represent a girl and \(B\) represent a boy. The sample space of two - children families is \(\{GG, GB, BG, BB\}\)
• \(P(A)=\frac{2}{4}=\frac{1}{2}\) (the cases \(\{GG, GB\}\))
• \(P(B)=\frac{1}{4}\) (the case \(\{GG\}\))
• \(P(A\cap B)=\frac{1}{4}\) (the case \(\{GG\}\))
• \(P(A)\times P(B)=\frac{1}{2}\times\frac{1}{4}=\frac{1}{8}

eq\frac{1}{4}=P(A\cap B)\)

• So events \(A\) and \(B\) are dependent.

Step4: Analyze Experiment 5

• For Event \(A\) (first roll is a \(5\)): \(P(A)=\frac{1}{6}\)
• For Event \(B\) (second roll is a \(2\)): \(P(B)=\frac{1}{6}\)
• Since the two rolls of the number cube are independent (the outcome of the first roll does not affect the outcome of the second roll), \(P(A\cap B)=\frac{1}{6}\times\frac{1}{6}=\frac{1}{36}\)
• \(P(A)\times P(B)=\frac{1}{6}\times\frac{1}{6}=\frac{1}{36}\)
• So \(P(A\cap B)=P(A)\times P(B)\), events \(A\) and \(B\) are independent.

Answer:

(c) Independent
(d) Dependent
(e) Independent