1) A die is selected at random from an urn that contains two...

1) A die is selected at random from an urn that contains two six-sided dice with the following characteristics. Each face of each die has one of the numbers 1, 2, 3, or 4. i. Die 1, has three 2s, and one 1, 3 and 4. ii. Die 2, has three 4s, and one 1, 2 and 3. iii. The first five rolls of the selected die yield the following in sequential order: 2, 3, 4, 1, and 4. What is the probability that the selected die is Die 2 given the observed sequence? 2) Suppose P(A) = 0.30, P(B) = 0.42, and P(A ? B) = 0.13. Calculate P( A | B^C ).

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00:01 In this question, we have, let's say, d1 is the event that we have a diet 1.

00:07 D2 is the events that we have diet 2.

00:12 And we just have two dires in the ur, and we select them randomly.

00:19 So we will have a probability one half, one half for each.

00:24 And then we have, so for die 1, let's consider the event first.

00:30 So we consider the probability that we choose that selected to start two given the observed sequence.

00:41 So we have d2 given the observed sequence, which is observed 2, 3, 4 ,14.

00:52 So using the conditional probability rule, we will have d2 and obs, 2, 3, 414.

01:03 Divided the probability that we observe 2, 3, 4, 1, 4.

01:08 Right.

01:11 And so for the first thing, the numerator will be equal to obs given d2 times probability of d2, right? and we just simplify that.

01:26 Obs 2, 3, 4, 1, 4.

01:27 It's just obs, okay? and for the denominator, we will have to use the complete probability formula, which is equal to probability of obs given d2 times pd2 plus probability of obs given d1 times probability of d1.

01:50 And then we need to determine for the observation of d2, if we have d2, it means that we have three -fourths.

01:59 This means that we have one half probability for getting a four and one of six for each of the other three values, one, two, and three...

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