Question

\or\ & \and\ probabilities for independent or exclusive events

1. events are independent if they do not affect each other. for example, if you flip a coin twice, the outcome of the first flip does not affect the second toss. are these events independent?

event a: rolling a 4 on the first fair die.
event b: rolling an even number on the 2nd die.
event a: it rains on a given day.
event b: you are wet when you arrive to school.

1. events are mutually exclusive if they cannot occur at the same time. example: when flipping a coin, the outcome can be either heads or tails, but not both simultaneously. are these events mutually exclusive?

event: drawing a card that is a heart or a 2
event: rolling a 3 or 6 on a 6 - sided die

Explanation:

Step1: Analyze independence for the first pair of die - rolling events

• The outcome of rolling the first die (getting a 4) has no impact on the outcome of rolling the second die (getting an even number). The probability of rolling an even number on the second die is \(\frac{3}{6}=\frac{1}{2}\) regardless of the result of the first die roll. So, they are independent.

Step2: Analyze independence for the rain - wetness events

• If it rains (\(P(A)>0\)), the probability that you are wet when you arrive at school (\(P(B|A)\)) is much higher than the probability that you are wet when it does not rain (\(P(B|

eg A)\)). Since \(P(B|A)
eq P(B)\), the events are not independent.

Step3: Analyze mutual - exclusivity for the card - drawing event

• There is a card (the 2 of hearts) that is both a heart and a 2. So, the two events (drawing a heart and drawing a 2) can occur simultaneously. Thus, they are not mutually exclusive.

Step4: Analyze mutual - exclusivity for the die - rolling event

• When rolling a 6 - sided die, you cannot get a 3 and a 6 at the same time. So, the events (rolling a 3 and rolling a 6) are mutually exclusive.

Answer:

1. First pair of events (Rolling a 4 on the first fair die and Rolling an even number on the 2nd die): Independent. Second pair of events (It rains on a given day and You are wet when you arrive to school): Not independent.
2. First event (Drawing a card that is a heart or a 2): Not mutually exclusive. Second event (Rolling a 3 or 6 on a 6 - sided die): Mutually exclusive.