Government Employees In 2013 about 66 % of full- time law...

Government Employees In 2013 about $66 \%$ of full- time law enforcement workers were sworn officers, and of those, $88.4 \%$ were male. Females however make up $60.7 \%$ of civilian employees. Choose one law enforcement worker at random and find the following. $$ \begin{array}{l}{\text { a. The probability that she is a female sworn officer }} \\ {\text { b. The probability that he is a male civilian employee }} \\ {\text { c. The probability that he or she is male or a civilian }} \\ {\text { employee }}\end{array} $$

Government Employees In 2013 about $66 \%$ of full- time law enforcement workers were sworn officers, and of those, $88.4 \%$ were male. Females however make up $60.7 \%$ of civilian employees. Choose one law enforcement worker at random and find the following.
$$
\begin{array}{l}{\text { a. The probability that she is a female sworn officer }} \\ {\text { b. The probability that he is a male civilian employee }} \\ {\text { c. The probability that he or she is male or a civilian }} \\ {\text { employee }}\end{array}
$$

Key Concepts

Joint Probability

Joint probability refers to the likelihood of two events occurring simultaneously. In problems where an individual has multiple attributes or belongs to more than one group, calculating the joint probability involves determining the intersection of these events. This concept is crucial for determining the probability of combined outcomes, such as someone being both a sworn officer and female.

Sample Space and Partitioning

The sample space in probability consists of all possible outcomes, and partitioning the sample space involves dividing it into distinct, non-overlapping groups. This is important for understanding and computing probabilities based on specific subgroups within a larger population, such as categorizing law enforcement workers by gender and employment type.

Union of Events and the Inclusion-Exclusion Principle

The union of events represents the probability that at least one of two events occurs. To compute this accurately, the inclusion-exclusion principle is used, where the probabilities of the individual events are added together and then the probability of their intersection is subtracted to avoid double counting. This is essential when considering cases like the probability of someone being male or a civilian employee.

Transcript

00:01 So this problem can appear pretty complex at first, but i'm going to introduce a method called a probability tree that can really help to break down and simplify this problem.

00:10 So we're given that 66 % of full -time law enforcement workers are sworn officers, and given that one is a sworn officer, 88 .4 % were male.

00:22 And we finally are given that females make up 60 .7 % of civilian employees.

00:30 And so now i want to take this information and organize it into a probability tree.

00:35 What i mean by this is the tree shows each stage, and by stage i mean each choice you have.

00:42 So the first choice is sworn slash civilian, and that's the type of officer.

00:48 And so you have one choice or the other choice with their corresponding probabilities.

00:52 And we were told that 66 % of full -time law enforcement workers are sworn officers.

00:57 So that's where this point -66 comes from.

00:59 The next stage of the tree is a gender representation.

01:03 So whether an officer is male or female are the two options in this problem.

01:08 And i also want to point out the next rule of a probability tree, are actually just all the rules of probability trees, because it could be a little confusing at first where you want to put that 88 .4 % and that 60 .7%.

01:22 Those are both conditional probabilities.

01:24 It's something i want to state about these branches right here.

01:27 Any number that is here is a conditional probability.

01:31 So, for instance, this value would be the probability that the officer is male given that they are a civil employee.

01:40 And so moving back up to placing this point 884, it goes here because the problem states, of those sworn officers, 88 .4 % were male.

01:50 So given that they are a sworn officer, what is the probability that you're male? this probability goes in this conditional branch right here of .884.

02:00 And the same goes for this .607, female employee given that they are a civil employee, also given by the problem.

02:09 The next information we can take from these trees to break them down is each set of branches like the .66 in this blank and this .884 in this blank and these two, these all add to one.

02:23 So what we can do from this .66.

02:28 Is 1 minus .66.

02:30 Sorry, 1 minus .66.

02:31 It's going to give us the probability right here of 0 .34.

02:35 Same with 1 minus .884 is going to be .116.

02:41 Gives us our probability right there.

02:43 And then 1 minus .607 gives us .393.

02:48 So now our tree is almost filled in.

02:51 The last stage of our tree that we need to fill in is right here.

02:54 These are called our intersection probability.

02:56 Are intersection probabilities, and these are our multiplication probabilities, this would be the probability that an officer is a sworn officer and male.

03:05 That probability is right here.

03:08 As similarly, we have sworn officer and female would go right here.

03:12 And so the paths multiply have written right here in the rules.

03:16 The paths of the tree, so 0 .66 times 0 .884 is going to give you the probability of both occurring, which in this case, when i plug it in, is 0 .4...

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