A coin is tossed three times and the sequence of heads and...

A coin is tossed three times and the sequence of heads and tails is recorded. a. List the sample space. b. List the elements that make up the following events: $(1) A=$ at least two heads, $(2) B=$ the first two tosses are heads, $(3) C=$ the last toss is a tail. c. List the elements of the following events: $(1) A^{c},(2) A \cap B,(3) A \cup C$

A coin is tossed three times and the sequence of heads and tails is recorded.
a. List the sample space.
b. List the elements that make up the following events: $(1) A=$ at least two heads, $(2) B=$ the first two tosses are heads, $(3) C=$ the last toss is a tail.
c. List the elements of the following events: $(1) A^{c},(2) A \cap B,(3) A \cup C$

Key Concepts

Intersection of Events

The intersection of two or more events is the set of outcomes that are common to all the events. This concept is crucial for determining the probability of multiple events occurring simultaneously and helps in understanding combined probabilities.

Complement of an Event

The complement of an event includes all outcomes in the sample space that do not satisfy the event's criteria. It is used to consider the scenario where the event does not occur and is key in applying the principle that the probability of an event and its complement sum to one.

Union of Events

The union of events comprises all outcomes that are present in at least one of the events. It is a fundamental concept used to determine the probability that any one of several conditions is met, by combining the probabilities of individual events while accounting for overlap.

Sample Space

The sample space is the complete set of all possible outcomes for a given experiment. In probability theory, particularly in experiments involving coin tosses or dice rolls, it provides the foundational listing from which events are defined and probabilities are calculated.

Event

An event represents a subset of the sample space and consists of outcomes that satisfy certain conditions. Events allow one to focus on and analyze specific outcomes within a broader set, facilitating probability calculations and logical reasoning.

Transcript

00:01 In question one, we have a probability situation in which we have a coin that is tossed three times in the sequence of heads and tails are recorded.

00:08 In part a, we want to know what is the sample space? so the sample space, all possibilities of heads and tails, i like to start with all heads and then move to one tails and just rearrange it to all the possible locations.

00:27 Then once we exhaust that, we're going to two tails and rearrange that one little heads and finish all.

00:39 With all tails.

00:41 So there's eight possible ways to flip a coin and you have your sequence of heads and tails here.

00:49 Part b, we want to list out a few different elements.

00:53 So number one here we want to list out a.

00:56 A is going to be at least two heads.

01:01 So going through the sample space, which is pretty easy because we have it above, list out all the different ways we have at least two heads.

01:10 So we have heads, heads, heads, heads, heads, heads, tails, heads, and tails, heads, heads.

01:22 So we have four different events.

01:25 And b, the question is, the first two talses are heads.

01:31 So this happened twice.

01:33 The very first one, all of them are heads.

01:34 And the next one, which were the first two were heads and the last one was tails.

01:41 And c is going to be the last toss is a tail.

01:45 So basically the last, the last one, tells, tells, tells, and then we have also two different events where heads, tells, tails, tails, heads, tails, and one at the beginning, heads, heads, tails...

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