Conditional probability

In probability, the conditional probability of an event is the probability that the event will occur, given that another event has already occurred.

For example, let's say we have a deck of cards and we want to know the probability of drawing an Ace of Spades, given that we have already drawn the Ace of Hearts. In this case, the probability of drawing the Ace of Spades is 1 out of 51, since there is only 1 Ace of Spades left in the deck and 51 total cards remaining.

We can express this probability using the following formula:

P(Ace of Spades | Ace of Hearts) = (Number of ways to get the Ace of Spades given that the Ace of Hearts has already been drawn) / (Total number of cards remaining)

So in this case, the probability of drawing the Ace of Spades given that the Ace of Hearts has already been drawn is 1/51.

It's important to note that the conditional probability of an event is not the same as the probability of the event occurring on its own. The probability of an event occurring on its own is called the unconditional probability.

Another expample (from the lesson)

Last semester, out of 170 students taking a particular statistics class, 71 students were “majoring” in social sciences and 53 students were majoring in pre-medical studies. There were 6 students who were majoring in both pre-medical studies and social sciences. What is the probability that a randomly chosen student is majoring in social sciences, given that s/he is majoring in pre-medical studies?

1 point

(71+53−6)/170

6/170

6/71

6/53

To solve this problem, we can use the formula for conditional probability, which is the probability of an event occurring given that another event has already occurred. The formula for conditional probability is:

P(A|B) = P(A and B) / P(B)

Where P(A|B) is the probability of event A occurring given that event B has already occurred, P(A and B) is the probability of both events A and B occurring, and P(B) is the probability of event B occurring.

In this problem, we are asked to find the probability that a student is majoring in social sciences given that they are majoring in pre-medical studies, so we can define event A as "majoring in social sciences" and event B as "majoring in pre-medical studies." We are given that there are 6 students who are majoring in both pre-medical studies and social sciences, so the probability of both events occurring is 6/170. We are also given that there are 53 students majoring in pre-medical studies, so the probability of event B occurring is 53/170. Plugging these values into the formula for conditional probability, we get:

P(A|B) = P(A and B) / P(B)

= (6/170) / (53/170)

= 6/53

So, the probability that a student is majoring in social sciences given that they are majoring in pre-medical studies is 6/53. The answer is therefore option 4: 6/53.