\(P(club\mid black)=\dfrac{P(club \: \cap\: black)}{P(black)}=\dfrac{0.25}{0.50}=0.50\) Given that a randomly selected card is black, there is a 50% chance that it's a club.
If events A and B are independent then \(P(A) = P(A \mid B)\). In other words, whether or not event B occurs does not change the probability of event A occurring.
A card is randomly drawn from a 52-card deck. Are the events of drawing an ace and drawing a heart independent? In a standard 52-card deck, there are 4 aces and 13 hearts. Therefore \(P(ace)=\frac{4}{52}\) and \(P(heart)=\frac{13}{52}\). Out of 13 hearts, 1 is an ace, which translates to \(P(ace \mid heart) = \frac{1}{13}\). To determine if these two events are independent we can compare \(P(A)\) to \(P(A\mid B)\). If we call being an ace event A and being a heart event B, then we're comparing \(P(ace)\) to \(P(ace \mid heart)\). \(P(ace)=\frac{4}{52}=0.0769\) \(P(ace \mid heart) = \frac{1}{13}=0.0769\) These values are identical, therefore we can conclude that the events of drawing an ace and drawing a heart are independent. |