1) a. Provide two independent events that you know the probability of.
b. Explain how you know these events are independent.
c. Find the probability that they both occur.
2) a. Provide two dependent events.
b. Explain how you know these events are dependent.
c. Explain why these dependent events might require a different method for calculating the probability of both events occurring.
1) a. Two independent events that are commonly known are flipping a fair coin and rolling a fair six-sided die.
b. These events are considered independent because the outcome of one event does not affect the outcome of the other event. Flipping a coin does not influence the result of rolling a die.
c. To find the probability of two independent events occurring, you can multiply the probabilities of each individual event. Let's consider the coin flip and die roll separately:
- The probability of getting heads on a fair coin flip is 1/2, or 0.5.
- The probability of rolling a specific number (let's say 4) on a fair six-sided die is 1/6.
To find the probability that both events occur, you multiply these probabilities:
P(coin flip = heads) x P(die roll = 4) = 0.5 x 1/6 = 1/12 ≈ 0.0833
So, the probability that both events occur is approximately 0.0833 or 8.33%.
2) a. Two dependent events can be choosing two cards from a standard deck without replacement.
b. These events are dependent because the outcome of the first event affects the probability of the second event. When you remove a card from the deck, the composition of the deck changes, altering the probabilities for the subsequent events.
c. Dependent events may require a different method to calculate the probability of both events occurring because the conditional probability plays a role. Conditional probability takes into account the given information or the outcome of the first event when calculating the probability of the second event.
For example, let's say you want to find the probability of drawing two red cards from a standard deck without replacement:
- The probability of drawing a red card as the first event is 26/52 since there are 26 red cards in a deck with 52 cards.
- If the first card drawn was red, there will be 25 red cards remaining in a deck with 51 cards.
- The probability of drawing another red card as the second event, given that the first card was red, is 25/51.
To calculate the probability of both events occurring, you multiply these conditional probabilities:
P(first card is red) x P(second card is red | first card is red) = 26/52 x 25/51 ≈ 0.2451
So, the probability of drawing two red cards from a standard deck without replacement is approximately 0.2451 or 24.51%.