Refer to Example 2.4.6. Calculate P[A2 |A], P[A3 |A] , and P[A4 |A] .

Name: Problem 40, Chapter 2: Probability, Statistics, and Queuing Theory with Computer Science Applications, Second Edition (Computer Science and Scientific Computing)
Uploaded: 2020-08-20T21:47:17-08:00
Duration: 3 min 46 s
Channel: Derrick Hanson
Description: Refer to Example 2.4.6. Calculate P[A2 |A], P[A3 |A] , and P[A4 |A] .

Refer to Example 2.4.6. Calculate P[A2 |A], P[A3 |A] , and ...

Key Concepts

Conditional Probability

This concept refers to the probability of an event occurring given that another event has already occurred. It is typically expressed as P(B|A), which equals P(A ? B) divided by P(A) when P(A) is greater than zero. Conditional probability is essential for problems that involve determining the likelihood of sub-events within a larger event.

Bayes' Theorem

Bayes' Theorem provides a method to update probability estimates as new evidence is obtained. It relates the conditional probability P(A|B) to P(B|A) along with the marginal probabilities P(A) and P(B). This theorem is especially useful in situations where the direction of conditioning needs to be reversed or clarified.

Law of Total Probability and Event Partitioning

The law of total probability allows for the calculation of the probability of an event by breaking it down into a sum over several disjoint sub-events or partitions of the sample space. When an event A is composed of several mutually exclusive events such as A2, A3, and A4, this law provides a systematic approach to compute overall probabilities from the probabilities of its parts.

Transcript

00:01 We are being asked to multiply three separate expressions together.

00:04 We have a monomial, we have a binomial, and another binomial.

00:08 Now, this is multiplication, and we know multiplication is commutative, meaning it doesn't really matter what order we decide to do this in.

00:17 It's up to you.

00:18 The important thing is that you do not try and distribute one of your expressions to multiple, meaning what you can't do is take 6p squared and distribute it to this expression of 3p minus 4 but then also try to take 6p squared and distribute it to this expression that doesn't work you can only distribute it to 1 you will choose okay now the order doesn't matter meaning you could take 6p squared times 3p minus 4 and do this part first and then multiply p plus 3 with whatever you get you could also take 6p 2 p squared and multiply it to p plus three and do that and then multiply three pot three p minus four to whatever you get from that you also could choose to foil three p minus four and p plus three together and then distribute six p squared into whatever you got from that all three of those options are acceptable and it truly does not matter which one you do you just have to make sure that you don't try and multiply one expression to multiple at the same time.

01:30 That's the only way that you end up messing yourself up.

01:35 Just for argument's sake, i'll go from left to right.

01:38 I mean, now i'm going to start with 6p squared and 3p minus 4.

01:42 So for now, we're just kind of ignoring p plus 3.

01:45 We're going to put in a box over there and leave it for a bit.

01:48 So 6p squared would need to be distributed into 3p minus 4.

01:52 6p squared times 3p, 6 times 3 is 18, p squared times p is p to the third.

02:00 6 p squared times negative 4, 6 times negative 4 is negative 24, and then p squared is basically along for the ride...