Two firms a and b consider bidding on a costume designing...

Two firms A and B consider bidding on a costume designing project for an upcoming new movie, which may or may not be awarded depending on the amounts of the bids. Firm A submits a bid and the probability is ¾ that it will get the project provided firm B does not bid. The probability is ¾ that B will bid, and if it does, the probability that A will get the project is only 1/3. If A gets the project, what’s the probability that B did not bid

Transcript

00:01 Here we're given several probabilities, and then we're asked to use them to determine a new conditional probability.

00:09 So we're given the probability that a wins, given b doesn't bid.

00:14 So when i draw the line over something, that means not b bids is 3 over 4.

00:21 The probability that b bids is also 3 over 4, and the probability that a wins when b bids or condition on the fact that b has bid is one -third.

00:34 And so now we want to find the probability that b does not bid condition on the event that a wins.

00:43 And so we're working with a bunch of conditional probabilities, and we're sort of switching the order.

00:48 So what we're going to want to use here is bay's theorem.

00:55 And what does bay's theorem say? well, i'll just give the specific instance.

01:00 It says that this probability is equal to the probability that a wins, given that b doesn't bid.

01:11 So we're switching the order here times the probability that b doesn't bid.

01:22 And then we're going to divide that by the probability that a wins.

01:31 Okay, so you'll see that we sort of have three different terms that we want to multiply and divide here.

01:39 And we know this one.

01:42 So this one we know.

01:44 This one, we don't quite know, although it's very easy to find out.

01:50 So maybe i'll just do that right now.

01:53 So the probability that b doesn't bid, one minus the probability that bids because b either bids or it doesn't.

02:05 So that's 1 over 4.

02:08 Okay, so the two probabilities on top are easy to figure out.

02:13 The one on bottom is a little bit harder, but we can use the law of total probability to figure it out.

02:21 So what are we doing here? well, the probability that a wins, that's going to be equal to the probability that a wins given that b bids times the probability that b bids plus, and maybe i'll go to a new line here, the probability that a wins given that b b doesn't.

02:50 Did times the probability that b doesn't bid...

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