State, with evidence, whether each of the following...

State, with evidence, whether each of the following statements is true or false: a. The complement of the union of two events is the intersection of their complements. b. The sum of the probabilities of collectively exhaustive events must equal $1 .$ c. The number of combinations of $x$ objects chosen from $n$ is equal to the number of combinations of $(n-x)$ objects chosen from $n$, where $1 \leq x \leq(n-1)$ d. If $A$ and $B$ are two events, the probability of $A$, given $B$, is the same as the probability of $B$, given $A$, if the probability of $A$ is the same as the probability of $B$. e. If an event and its complement are equally likely to occur, the probability of that event must be $0.5$. f. If $A$ and $B$ are independent, then $\bar{A}$ and $\bar{B}$ must be independent. g. If $A$ and $B$ are mutually exclusive, then $\bar{A}$ and $\bar{B}$ must be mutually exclusive.

State, with evidence, whether each of the following statements is true or false:
a. The complement of the union of two events is the intersection of their complements.
b. The sum of the probabilities of collectively exhaustive events must equal $1 .$
c. The number of combinations of $x$ objects chosen from $n$ is equal to the number of combinations of $(n-x)$ objects chosen from $n$, where $1 \leq x \leq(n-1)$
d. If $A$ and $B$ are two events, the probability of $A$, given $B$, is the same as the probability of $B$, given $A$, if the probability of $A$ is the same as the probability of $B$.
e. If an event and its complement are equally likely to occur, the probability of that event must be $0.5$.
f. If $A$ and $B$ are independent, then $\bar{A}$ and $\bar{B}$ must be independent.
g. If $A$ and $B$ are mutually exclusive, then $\bar{A}$ and $\bar{B}$ must be mutually exclusive.

Key Concepts

De Morgan's Laws

De Morgan's Laws relate the complement of unions and intersections in set theory and probability. They state that the complement of the union of two sets is the intersection of their complements, and vice versa, which is a fundamental concept in probability for transforming expressions involving complements.

Collectively Exhaustive Events

Collectively exhaustive events are a set of events that together cover the entire sample space, meaning that at least one of the events must occur. This concept underlies one of the basic axioms of probability, wherein the sum of the probabilities of collectively exhaustive events is equal to 1.

Binomial Coefficient Symmetry

Binomial coefficient symmetry is the combinatorial property that states the number of ways to choose x objects from n objects is the same as the number of ways to choose (n - x) objects from n, often expressed as C(n, x) = C(n, n - x). This symmetry is a key element in combinatorics and is widely used in probability distributions and counting arguments.

Conditional Probability

Conditional probability measures the probability of an event occurring given that another event has occurred. It is defined as P(A|B) = P(A ? B) / P(B), assuming P(B) > 0. This concept is crucial in understanding how the presence of one event can affect the likelihood of another, though it is not generally symmetric.

Complementary Events

Complementary events are pairs of events where one event's occurrence means the other's non-occurrence, and together they exhaust the entire sample space. Their probabilities add up to 1, making this concept central to understanding basic probability calculations and the idea that an event and its complement are mutually exclusive and exhaustive.

Independence of Events

Independence of events refers to a situation where the occurrence of one event does not affect the probability of the other. When events are independent, the probability of their intersection is the product of their probabilities. This concept extends to certain operations on events, such as the independence between the complements of independent events.

Mutually Exclusive Events

Mutually exclusive events are events that cannot occur simultaneously; if one occurs, the other cannot. This concept is critical in probability theory for understanding how events interact, particularly in contrast to independent events. It is important to note that mutual exclusivity and independence are distinct concepts.

Transcript

00:01 So before we start, let's take a diagram and here's a sample space and let's say this is event a and this is event b.

00:09 And we can see that the intersection is right in here.

00:16 That is the probability of a intersect b and the union is the probability of all these.

00:23 So let's look at the questions.

00:25 And the first one states that the probability of the union, cannot be less than the probability of the intersection.

00:34 So the probability of the union cannot be less than the probability of their intersection.

00:46 And that is true.

00:48 The union has to be at least the size of the intersection if the two were exactly the same set.

00:54 So that answer is definitely true.

00:57 Part b, we have that the probability of the union, union cannot be more than the sum of their individual probabilities.

01:08 And we know that the probability of a union b is equal to the probability of a plus the probability of b.

01:16 And then, so it can't be more than that, because then we subtract away the probability of their intersection.

01:25 So when it states that the probability of the union cannot be more than the sum of the individual probabilities, that is true.

01:38 It can't be more than that because the sum of the probabilities is the most it could ever be, and it can be smaller if there's intersection.

01:45 So that is true.

01:48 Part c.

01:50 Part c says that the probability of the intersection cannot be greater than either of the two individual probabilities.

01:59 So the probability of the intersection, probability of that intersection, cannot be greater.

02:05 Than either one of these two probabilities.

02:08 And that is true, because the probability of the intersection is part of each of these events.

02:14 And so it can't be bigger than the events themselves.

02:20 Question d.

02:23 Question d.

02:25 We have an event and its complement are mutually exclusive...

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