Question

independent events:
(with replacement)
$p(a\\ and\\ b)=p(a)\times p(b)$;
p(a and b) means both events happened
$p(a|b)=p(a)$ (b didnt affect p(a))
$p(b|a)=p(b)$ (a didnt affect p(b))
p(a|b) means the probability of a given b
or p(a) assuming b already happened
dependent events:
(without replacement)
$p(a\\ and\\ b)\
eq p(a)\times p(b)$
$p(a|b)\
eq p(a)$;
event b affects the chances of event a
happening
$p(b|a)\
eq p(b)$;
event a affects the chances of event b
happening
test for independence:

1. multiply p(a) times p(b)
2. if the answer equals p(a and b), independent because $p(a\\ and\\ b)=p(a)\times p(b)$
3. if the answer is anything else, dependent because $p(a\\ and\\ b)\

eq p(a)\times p(b)$
determine whether the scenario involves independent or dependent events.

1. you flip a coin and then roll a fair

six - sided die. the coin lands heads - up
and the die shows a one.

1. a bag contains eight red marbles and four

blue marbles. you randomly pick a
marble and then pick a second marble
without returning the marbles to the bag.
the first marble is red and the second
marble is blue.

1. a box of chocolates contains five milk

chocolates, five dark chocolates, and five
white chocolates. you randomly select
and eat three chocolates. the first piece is
milk chocolate, the second is dark
chocolate, and the third is white chocolate.

1. a cooler contains ten bottles of sports

drink: four lemon - lime flavored, three
orange flavored, and three fruit - punch
flavored. three times, you randomly grab
a bottle, return the bottle to the cooler, and
then mix up the bottles. the first time,
you get a lemon - lime drink. the second
and third times, you get fruit - punch.

1. $p(a)=0.4$, $p(b)=0.7$, $p(a\\ \\&\\ b)=0.28$
2. $p(a)=0.5$, $p(b)=0.43$, $p(a\\ \\&\\ b)=0.186$
3. $p(a)=0.13$, $p(b)=0.15$, $p(a\\ \\&\\ b)=0.0195$
4. $p(a)=0.37$, $p(b)=0.41$, $p(a\\ \\&\\ b)=0.2351$
5. $p(a)=0.2$, $p(b)=0.8$, $p(a\\ \\&\\ b)=0.28$
6. $p(a)=0.3$, $p(b)=0.95$, $p(a\\ \\&\\ b)=0.285$

Explanation:

Step1: Analyze each scenario

• Scenario 1: Flipping a coin and rolling a die. The outcome of the coin - flip does not affect the outcome of the die - roll.
• Scenario 2: Picking marbles without replacement. The first pick affects the composition of the bag for the second pick.
• Scenario 3: Selecting and eating chocolates (without replacement as you eat them). The first selection affects the available chocolates for the next selections.
• Scenario 4: Grabbing a bottle and then returning it. Since the bottle is returned, the first grab does not affect the subsequent grabs.
• Scenario 5: Calculate \(P(A)\times P(B)=0.4\times0.7 = 0.28\). Since \(P(A\cap B)=0.28\), by the test for independence \(P(A\cap B)=P(A)\times P(B)\), the events are independent.
• Scenario 6: Calculate \(P(A)\times P(B)=0.5\times0.43=0.215\). Since \(P(A\cap B) = 0.186

eq0.215\), the events are dependent.

• Scenario 7: Calculate \(P(A)\times P(B)=0.13\times0.15 = 0.0195\). Since \(P(A\cap B)=0.0195\), by the test for independence \(P(A\cap B)=P(A)\times P(B)\), the events are independent.
• Scenario 8: Calculate \(P(A)\times P(B)=0.37\times0.41=0.1517\). Since \(P(A\cap B)=0.2351

eq0.1517\), the events are dependent.

• Scenario 9: Calculate \(P(A)\times P(B)=0.2\times0.8 = 0.16\). Since \(P(A\cap B)=0.28

eq0.16\), the events are dependent.

• Scenario 10: Calculate \(P(A)\times P(B)=0.3\times0.95 = 0.285\). Since \(P(A\cap B)=0.285\), by the test for independence \(P(A\cap B)=P(A)\times P(B)\), the events are independent.

Answer:

1. Independent; 2. Dependent; 3. Dependent; 4. Independent; 5. Independent; 6. Dependent; 7. Independent; 8. Dependent; 9. Dependent; 10. Independent