Draw a card from a single deck. The probability that the card will be a Jack is 4/52. But suppose you get a glimpse of the card, enough to know that it is a face-card. Now things are different. You have more information about the card. The probability that the card is a Jack "given that it is a face-card" is called a conditional probability.We know there are only 12 face-cards (4 Jacks, 4 Queens and 4 Kings) in a single deck, so the probability that the card is a Jack "given that it is a face-card now becomes 4/12, or 1/3.
In general terms, the "conditional probability" P[A|B] is the probability that A occurs, given that B has occurred and is formally defined as:
P[A|B] = P[A and B] / P[B] The notation P[A|B] is special shorthand that means "the probability of A, given that we know B has occurred." To obtain the answer for P[Jack|Face-card], we translate it into a ratio of the chances of both A and B happening to the chances of B happening. In other words, the conditional probability for our example is given by the following equation:
P[Jack|Face-card] = P[of being a Jack and also a face-card] / P[of being a face-card]
So we get P[Jack|Face-card] = (4/52)/(12/52) = 1/3
Conditional probabilities can be confusing. You must be clear on what P[A|B] really means. I repeat, P[A|B] means "the probability of A occurring, given that B has occurred."
Example 1:
Let's try working through an example. "Given that you have been dealt an Ace as your first card, what is the probability of being dealt a blackjack?" Using the formula for condtional probability, we get:P[Blackjack|Ace has been dealt] = P[Blackjack and Ace has been dealt]/P[Ace has been dealt]
For independent events, P[A and B] = P[A] * P[B]. The probability of receiving a blackjack is (4/52)*(16/51) + (16/52)(4/51). But because we are specifying that the Ace must come first for our Blackjack, we must not include any blackjacks where the 10-valued card comes first. Therefore, the blackjack is made up of an Ace first and only then a 10-value card, so we can say that P[Blackjack and Ace has been dealt first] = (4/52 * 16/51). Therefore, P[Blackjack|Ace has been dealt first] = P[Blackjack and Ace has been dealt first]/P[Ace has been dealt first] = (4/52 * 16/51)/(4/52) = 16/51.
Example 2:
This time, let's say we have been dealt a King. Now, "What is the probability of obtaining a Blackjack?" If you have worked through Example 1, then you may be able to directly notice that P[Blackjack|King] = P[Blackjack and King]/P[King] = (4/52 * 4/51)/4/52 = 4/51.If not, then notice that although the probability of a blackjack is calculated as (4/52)*(16/51) + (16/52)(4/51), we must not include any blackjacks where cards other than a king come first. There are four kings and four aces. Therefore, the probability of a king-first blackjack is P[Blackjack and King] = P[King] * P[Ace] = (4/52) * (4/51). To get the final answer, we have to divide by P[King] and we know that P[King] = 4/52. So, P[Blackjack|King] = P[Blackjack and King]/P[King] = (4/52 * 4/51)/4/52 = 4/51.