Which Conditional Probabilities Are Correct? Check All That Apply
Conditional probabilities are a fundamental concept in probability theory that allow us to determine the likelihood of an event occurring given that another event has already happened. In this article, we will explore some common scenarios and discuss which conditional probabilities are correct. To ensure clarity, we will check all the probabilities that apply in each scenario.
Scenario 1: Two Fair Coins Are Tossed
Consider the scenario where two fair coins are tossed. Let A be the event of getting at least one head, and B be the event of getting at least one tail. We need to determine the conditional probabilities P(A|B) and P(B|A).
To calculate P(A|B), we first need to find the probability of event A, which is the probability of getting at least one head. This can be calculated as follows:
P(A) = P(head on the first coin) + P(head on the second coin) – P(head on both coins)
P(A) = 1/2 + 1/2 – 1/4 = 3/4
Next, we need to find the probability of event B, which is the probability of getting at least one tail. This can be calculated as follows:
P(B) = P(tail on the first coin) + P(tail on the second coin) – P(tail on both coins)
P(B) = 1/2 + 1/2 – 1/4 = 3/4
Now, we can calculate the conditional probabilities:
P(A|B) = P(A and B) / P(B)
P(A|B) = (1/4) / (3/4) = 1/3
P(B|A) = P(A and B) / P(A)
P(B|A) = (1/4) / (3/4) = 1/3
Therefore, both P(A|B) and P(B|A) are equal to 1/3. Thus, both conditional probabilities are correct.
Scenario 2: A Card Is Drawn from a Standard Deck
In this scenario, let A be the event of drawing a face card (Jack, Queen, or King), and B be the event of drawing a red card. We need to determine the conditional probabilities P(A|B) and P(B|A).
The probability of event A, which is the probability of drawing a face card, is:
P(A) = (3/4) (3/4) = 9/16
The probability of event B, which is the probability of drawing a red card, is:
P(B) = (1/2) (3/4) = 3/8
Now, we can calculate the conditional probabilities:
P(A|B) = P(A and B) / P(B)
P(A|B) = (1/4) / (3/8) = 2/3
P(B|A) = P(A and B) / P(A)
P(B|A) = (1/4) / (9/16) = 4/9
In this case, P(A|B) = 2/3 and P(B|A) = 4/9. Both conditional probabilities are correct.
In conclusion, when determining which conditional probabilities are correct, it is essential to consider the specific scenario and calculate the probabilities accordingly. By following the steps outlined in this article, you can ensure that the conditional probabilities you calculate are accurate.
