1a. Two independent events that I know the probability of are:
- Rolling a fair six-sided die and getting a 4 (Probability = 1/6)
- Flipping a fair coin and getting heads (Probability = 1/2)
1b. I know these events are independent because the outcome of one event does not affect the outcome of the other event. In other words, the probability of getting a 4 on a die roll is unaffected by whether the coin flip results in heads or tails, and vice versa.
1c. The probability that both events occur is found by multiplying the probabilities of each event:
\[ P(\text{Die rolls a 4 and Coin flips heads}) = P(\text{Die rolls a 4}) \times P(\text{Coin flips heads}) = \frac{1}{6} \times \frac{1}{2} = \frac{1}{12} \]
2a. Two dependent events are:
- Drawing a red card from a standard deck of 52 playing cards, without replacement.
- Drawing a second red card from the remaining deck, without replacement.
2b. I know these events are dependent because the outcome of the first event affects the probability of the second event. After drawing a red card from the deck, there are fewer red cards remaining in the deck, which changes the probability of drawing a second red card.
2c. These dependent events might require a different method for calculating the probability of both events occurring because the probability of the second event depends on the outcome of the first event. To calculate the probability of both events occurring, we need to multiply the conditional probabilities of each event given the outcome of the previous event. This involves adjusting the probabilities for the reduced sample space after the first event has occurred.
