You roll a fair four sided die where the sides are labeled 1...

You roll a fair four-sided die, where the sides are labeled 1, 2, 3, and 4. If the result is 1 or 2, you roll the die once more; otherwise, you do not. Note that you roll the die at most twice (so if you get a 2, and then another 2, you do not roll a third time). Consider: B = "the sum total of your rolls is at least 4". (a) Draw a suitable tree diagram for this game. (b) What is the probability of the event B? (c) Given that the sum total of your rolls is at least 4, what is the probability that at the 1st roll you got a 2?

Transcript

00:01 In this question, we're given that a fair four -sided die is robed, and the sides are labelled 1, 2, 3 and 4.

00:07 If the result in the first row is 1 or 2, the die will be rolled one more time, and that's it.

00:15 Now, event b is the event where the sum total of the row is at least 4.

00:21 In part a, we have to draw the 3 diagram.

00:24 So we have, in the first row, possible is 1, 2, 3 or 4.

00:32 So we have the branches like this.

00:36 Now in each branch, of course, since the dye is fair, it's one quarter chance.

00:45 Now in the second row, it only happens if the first row is chose one or two.

00:52 So if it is one, you have four possibility.

00:58 One, two, three, four.

01:01 If it is two, you also have four possibility.

01:09 And for the first row, three and four, there will.

01:12 Will not be a second rule.

01:14 So in each of the branch over here, the probability of each outcome is also one quarter, similar for this branch as well.

01:32 Okay, in part b, we want to find the probability of the event b, which is the probability that the sum total of the rows is at least four.

01:42 So it's either probability of the first row is one, second row will be three, or all means plus, the first row is one, second row is four, or the first row is two, second row is two, or first row is two, second row is three, or first row is two, second row is three, or first row is two, second row is two, second, row is 4 or the first row is 4 and there is no second rule.

02:26 So the first possibility here of 1 3 would just be taking 1 3 here, taking 1 quarter times 1 quarter.

02:40 So you can see for every pair actually is 1 quarter times 1 quarter.

02:43 So let's look here.

02:47 So it's 1 quarter times 1 quarter plus for this pair is also one quarter times one quarter all the way to this pair is one quarter times one quarter plus now probability of the first row getting the number four is just one quarter so you can see that there is five of the one quarter times one quarter so it's five out of sixteen plus one quarter and so the final answer is zero point five six two five okay, in part c, we want to find the probability that the first row is a two, given that the sum total of the rows is at least four...

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