An urn contain b black balls and w white balls sequentially...

An urn contains b black balls and w white balls. Sequentially remove a random ball from the urn until none is left. Which of the following observed color sequences would you think is more likely: first all white balls then all black balls or alternating white and black balls until one color is exhausted, then the other color until it is exhausted? For b=4 and w=2, calculate the probability of: (i) (white, white, black, black, black, black) (ii) (white, black, white, black, black, black)

Transcript

00:01 All right, for this question, we're told that there's an urn containing four black balls and two white balls.

00:05 And we're drawing them out in order until they're all gone.

00:08 And we're asked, what should we think is more likely, the event that we draw all the white first and then all the black, or that we alternate white with black until white runs out and then black for the rest.

00:18 So again, we're finding the probability that we get white, white, then black, black, black, black.

00:23 And then we're finding the probability white, black, black, black.

00:26 Black black.

00:27 And so generally speaking, when we have consecutive trials like this, the probability of any given outcome or order matters is going to be identical, no matter what the order of events is actually happening.

00:43 And so let's show that these probabilities are going to be the same.

00:47 So what is the probability that the first ball that we draw is white? well, there's two white balls out of six total.

00:53 So this is going to be two out of six.

00:55 Then what is a probability? probability that with one white ball gone, we draw another white ball.

01:01 Well, there's one white ball left times a total of five balls that are left now.

01:08 All right.

01:08 So then what is the probability of drawing a black ball? when all the white balls are gone, well, that's going to be 100%.

01:17 Or saying that there are four black balls out of four total balls.

01:24 And then we're going to continue with this.

01:25 There's three black balls out of three total balls all the way until we run out and get a six ball.

01:29 And so this just gives us a probability of one -third times one -fifth or one -fifteenth, which is equal to 0 .0666 repeating or 0 .0667...

Question

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