Assume that ℙ(A)=0.5, ℙ(A ∩C)=0.2, ℙ(C)=0.4, ℙ(B)=0.4, ℙ(A ∩B ∩C)=0.1, ℙ(B ∩C)=0.2, and ℙ(A ∩B)=

Assume that ℙ(A)=0.5, ℙ(A ∩C)=0.2, ℙ(C)=0.4, ℙ(B)=0.4, ℙ(A...

Assume that $\mathbb{P}(A)=0.5, \mathbb{P}(A \cap C)=0.2, \mathbb{P}(C)=0.4, \mathbb{P}(B)=0.4, \mathbb{P}(A \cap B \cap C)=0.1$, $\mathbb{P}(B \cap C)=0.2$, and $\mathbb{P}(A \cap B)=0.2$. Calculate the following probabilities: (a) $\mathbb{P}(A \cup B \cup C)$ (b) $\mathbb{P}\left(A^c \cap(B \cup C)\right)$ (c) $\mathbb{P}\left((B \cap C)^c \cup(A \cap B)^c\right)$ (d) $\mathbb{P}(A)-\mathbb{P}(A \cap C)$

Assume that $\mathbb{P}(A)=0.5, \mathbb{P}(A \cap C)=0.2, \mathbb{P}(C)=0.4, \mathbb{P}(B)=0.4, \mathbb{P}(A \cap B \cap C)=0.1$, $\mathbb{P}(B \cap C)=0.2$, and $\mathbb{P}(A \cap B)=0.2$. Calculate the following probabilities:
(a) $\mathbb{P}(A \cup B \cup C)$
(b) $\mathbb{P}\left(A^c \cap(B \cup C)\right)$
(c) $\mathbb{P}\left((B \cap C)^c \cup(A \cap B)^c\right)$
(d) $\mathbb{P}(A)-\mathbb{P}(A \cap C)$

Key Concepts

Complement Rule

The complement rule states that the probability of an event not occurring is equal to one minus the probability of the event occurring. This rule is particularly helpful in problems where it is easier to calculate the probability of the complement rather than the event itself. It is a direct consequence of the basic axiom that the total probability of all outcomes in a sample space is one.

Inclusion-Exclusion Principle

This principle is used to calculate the probability of the union of several events. It involves summing the probabilities of individual events, then subtracting the probabilities of all pairwise intersections, and finally adding back the probabilities of the triple (or higher order) intersections to avoid overcounting. This method is essential when events overlap, as it ensures that each outcome is counted exactly once.

Set Operations in Probability

Understanding and manipulating set operations such as unions, intersections, and complements is fundamental in probability theory. The union of events represents the occurrence of at least one event, the intersection represents the simultaneous occurrence of multiple events, and the complement represents the non-occurrence of an event. These operations allow for the formulation and simplification of complex probability expressions.

Transcript

00:01 So we have a whole bunch of probabilities given to us and we want to find some answers.

00:05 And let's draw a venn diagram.

00:08 This is going to be the easiest way to make this a visual.

00:11 So we have event a, event b, and we have event c.

00:18 And let's put these probabilities in here.

00:21 So we know that the intersection of all three is .1.

00:28 We know that the probability that a intersects c is .2.

00:33 So this intersection is .2, so this must be .1 because these two have to add to .2.

00:38 The intersection of b and c is .2.

00:43 B and c is .2, so this must also be .1.

00:47 And the intersection of a and b is .2, so this must be a .1.

00:54 So now the sum in a has a probability of .5, so .1, .1, .1, .3, so this part must be .2 for those to add up to .5.

01:05 Now the probability of c has to be .4, and so .1, .1, .1, this must be .1.

01:13 And for b is also the same story, this must be .1.

01:20 So let's add all this up.

01:21 .2, .1, so we know that this adds up to .5, .6, .7, .8, so this must be .2 outside.

01:31 So now let's answer our question.

01:33 The probability of a union b union c is everything included in those, and that is .8.

01:43 B we want to find the probability of a complement intersected with b union c.

01:51 So we want the intersection of everything outside of a, but we want it to be united with b and c...

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