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Independence in Probability
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Flipping a coin and rolling a die.
These events are independent since the result of the coin flip does not affect the outcome of the die roll.
Filling out a multiple-choice test by guessing all answers randomly.
Each question guessed is an independent event since previous guesses do not impact future ones.
Drawing two cards from a deck without replacement.
These events are dependent since the outcome of the first draw affects the probabilities of the second draw.
Selecting a random student being left-handed and having blue eyes.
These events could be independent if being left-handed is not related to having blue eyes, but verifying requires empirical data.
Simultaneously drawing two cards from two different decks.
These events are independent because the cards are being drawn from separate decks, not affecting each other's outcomes.
Rainfall on two consecutive days.
This is potentially a dependent event, as the weather on one day can be influenced by the previous day's conditions.
Being dealt an ace first, and then a king from a standard deck of cards without replacement.
These events are dependent because removing an ace changes the composition of the deck for the second draw.
The birthday of one person and the birthday of another person chosen at random.
These events are independent unless there is some reason to believe that the choice of one person depends on the birthday of another.
The occurrence of an earthquake on two separate tectonic plates within the same hour.
These events may be dependent or independent, depending on whether or not the plates interact or are related.
Getting two heads in two consecutive flips of the same coin.
These events are independent; the outcome of the first flip does not influence the outcome of the second flip.
Two individuals from different cities getting the same exact score on a standardized test.
These events are independent, assuming the individuals did not collude or share answers.
Pulling a red ball from a bag of multicolored balls, replacing it, and then pulling another ball.
These events are independent because the replacement restores the original conditions after each pull.
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