Which events are independent?
You choose 2 different ice cream flavors.
You study English 20 min nightly, then you get an A on the next test.
You draw a card from a deck, replace it and draw a second.
You draw a card and don't replace it. Then you draw another.
Choosing 2 different ice cream flavors and drawing a card from a deck, replacing it and drawing a second card are independent events. Studying English 20 min nightly and getting an A on the next test, and drawing a card and not replacing it before drawing another are not independent events.
You choose 2 different ice cream flavors.
This is an independent event. The choice of one ice cream flavor does not affect or influence the choice of the second ice cream flavor. The two events occur separately and have no effect on each other.
To determine whether events are independent, we need to consider whether the occurrence of one event affects the probability of the other event happening. If the occurrence of one event does not affect the probability of the other event, then the events are independent.
Let's analyze each scenario to see if the events are independent:
1. Choosing 2 different ice cream flavors:
This event is independent because the flavor of the first ice cream does not affect the probability of choosing a different flavor for the second ice cream. Each selection is made separately, without any influence on the other.
2. Studying English for 20 minutes nightly and getting an A on the next test:
These events are likely not independent because studying for a specific amount of time each night can have a direct impact on your performance on the test. The more you study, the higher the probability of getting a good grade. Thus, these events are dependent.
3. Drawing a card from a deck, replacing it, and then drawing a second card:
This scenario includes two independent events because replacing the first card ensures that each card drawn has an equal probability of being chosen. The probability distribution remains the same after each draw, making the events independent.
4. Drawing a card, not replacing it, and drawing another card:
In this scenario, the events are dependent. After the first card is drawn and not replaced, the probability distribution changes because the number of cards in the deck decreases. So, the first draw affects the probability of the second draw, making the events dependent.
In summary, the events that are independent are choosing 2 different ice cream flavors and drawing a card from a deck, replacing it, and then drawing a second card.