A coin is to be tossed four times. Define events X and Y...

A coin is to be tossed four times. Define events $X$ and $Y$ such that $X:$ first and last coins have opposite faces $Y:$ exactly two heads appear Assume that each of the sixteen Head/Tail sequences has the same probability. Evaluate (a) $P\left(X^{C} \cap Y\right)$ (b) $P\left(X \cap Y^{C}\right)$

A coin is to be tossed four times. Define events $X$ and $Y$ such that
$X:$ first and last coins have opposite faces
$Y:$ exactly two heads appear
Assume that each of the sixteen Head/Tail sequences has the same probability.
Evaluate
(a) $P\left(X^{C} \cap Y\right)$
(b) $P\left(X \cap Y^{C}\right)$

Key Concepts

Sample Space

The sample space in probability theory is the set of all possible outcomes of a random experiment. In experiments like coin tossing, where each toss can result in one of two outcomes, the total number of outcomes is determined by raising the number of outcomes per trial to the power of the number of trials. This forms the basis for solving probability problems because it defines the universe over which events are evaluated.

Event Definitions

Events in probability are specific sets of outcomes within the sample space that satisfy certain criteria. They are defined by particular conditions or properties that the outcomes must meet. For instance, conditions can involve the outcome of specific trials or the total number of occurrences of an event, and they form the basic building blocks for calculating probabilities.

Uniform Probability Model

A uniform probability model assumes that all outcomes in the sample space are equally likely. This assumption simplifies the process of calculating probabilities, as the chance of any event occurring can be computed by taking the ratio of the number of favorable outcomes to the total number of outcomes. It is a common model used in many introductory probability problems.

Combinatorial Analysis

Combinatorial analysis involves counting the number of ways certain conditions can be fulfilled within the sample space. It uses techniques from counting and combinatorics, such as combinations and permutations, which are essential when calculating probabilities for events defined by exact counts or specific placements of outcomes.

Probability Operations (Intersection and Complement)

Operations such as intersection and complement are crucial in probability theory. The intersection of two events includes only those outcomes that satisfy both events simultaneously, whereas the complement of an event includes all outcomes that do not satisfy it. These operations allow for the manipulation and combination of events in order to solve more complex probability problems.

Transcript

00:01 In this problem it is said that a coin is tossed four times.

00:04 We have been given two events x and y and we need to find certain probabilities assuming that each of the 16 head tail sequences have the same probability.

00:14 So first of all let us draw a tree diagram to determine the 16 head tail sequences.

00:19 So we toss the coin once it either lands on heads or tails.

00:25 On the second toss it either lands on heads or tails.

00:29 So the second toss heads or tails.

00:33 And then on the third toss it either lands on heads or tails.

00:41 And then on the fourth toss it either lands on heads or tails.

00:52 Now using this tree diagram we can determine the 16 head tail sequences.

00:57 The first one will be h h h h then we'll have h h h t then we'll have h h t h and so on.

01:05 So let's just write down all of those head tail sequences.

01:23 Now we can find the required probabilities.

01:26 So first of all we want to find the probability of x complement intersection y.

01:31 So this will be the number of favorable outcomes divided by the total number of outcomes.

01:35 The total number of outcomes is 16.

01:37 And what will the favorable outcomes be? so we have x complement intersection y.

01:42 Intersection represents and.

01:44 So x is the event that the first and last coins have opposite faces.

01:49 So x complement means that the first and last coins have the same face...

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