A jar contains black and white marbles. Two marbles are chosen without replacement. The probability of selecting a black marble and then a white marble is 0.34, and the probability of selecting a black marble on the first draw is 0.47. What is the probability of selecting a white marble on the second draw, given that the first marble drawn was black? the source
It is the English of this problem that causes me problems.
The probability of selecting a black marble and then a white marble is 0.34
Shouldn't that be P(White | Black) not P(Black and White) because P(Black and White) = P(White and Black). $P(B \cap W) = P(W \cap B)$ The reason I think this is because
The probability of selecting a black marble and then a white marble is 0.34
The site says that means P(Black and White). I would agree with that if then was removed from the sentence.
Conditional Probability word problems always confuse me because the wording is always non-standard.
Is there a standard grammatical syntax for conditional probability problems to distinguish between P(A and B) and P(A | B).
I understand P(A and B) means that both events occur while P(A | B) means that given (knowing) event B occurred what is the probability that event A occurs(follows)
Best Answer
The events labelled "Black" and "White" are really "first marble black" and "second marble white" (I think this is confusing: it would be better to label these as something like "B1" and "W2"). If they just said "a black marble and a white marble" it would also include the case of a white marble first and a black marble second.