(a) Three typed letters and their envelopes are piled on a...

(a) Three typed letters and their envelopes are piled on a desk. If someone puts the letters into the envelopes at random (one letter in each), what is the probability that each letter, gets into its own envelope? Call the envelopes $A, B, C$, and the corresponding letten a, $b, c$, and set up the sample space. Note that " $a$ in $C, b$ in $B, c$ in $A^{\prime \prime}$ is ome point in the, sample space. (b) What is the probability that at least one letter gets into its own envelope? Hiwt: What is the probability that no letter gets into its own envelope? (c) Let $A$ mean that a got into envelope $A$, and so on. Find the probability $P(A)$ that a got into $A$. Find $P(B)$ and $P(C)$. Find the probability $P(A+B)$ that either $a$ or $b$ or both got into their correct envelopes, and the probability $P(A B)$ that both got into their correct envelopes. Verify cquation (3.6).

(a) Three typed letters and their envelopes are piled on a desk. If someone puts the letters into the envelopes at random (one letter in each), what is the probability that each letter, gets into its own envelope? Call the envelopes $A, B, C$, and the corresponding letten a, $b, c$, and set up the sample space. Note that " $a$ in $C, b$ in $B, c$ in $A^{\prime \prime}$ is ome point in the, sample space.
(b) What is the probability that at least one letter gets into its own envelope? Hiwt: What is the probability that no letter gets into its own envelope?
(c) Let $A$ mean that a got into envelope $A$, and so on. Find the probability $P(A)$ that a got into $A$. Find $P(B)$ and $P(C)$. Find the probability $P(A+B)$ that either $a$ or $b$ or both got into their correct envelopes, and the probability $P(A B)$ that both got into their correct envelopes. Verify cquation (3.6).

Key Concepts

Events and Their Intersections

In probability, events are outcomes or sets of outcomes from the sample space, and intersections of events represent scenarios where multiple conditions happen together. Analyzing these intersections, such as multiple items being in their correct positions, is essential for accurately calculating joint and individual probabilities in complex assignment problems.

Complement Rule

The complement rule is a fundamental principle in probability which states that the probability of an event occurring is one minus the probability of it not occurring. This approach is particularly useful in calculations involving 'at least one' occurrences, as it can be simpler to count the non-occurrence and subtract from one.

Fixed Points in Permutations

A fixed point in a permutation is an element that remains in its original position after the rearrangement. Determining the number of fixed points and their distribution is an essential aspect of problems where matches or correct placements are of interest, and it is often used in conjunction with the concept of derangements.

Permutations

A permutation is an ordered arrangement of items. In problems involving random assignment, such as matching items to positions, the total number of outcomes is determined by the number of permutations of the items. This concept is the basis for counting the possible arrangements and underpinning probability calculations.

Sample Space

The sample space is the collection of all possible outcomes of a random experiment. In counting problems, especially those involving random assignments or arrangements, listing or characterizing the sample space helps in determining the total number of equally likely outcomes, which is essential for calculating probabilities.

Derangements

A derangement is a specific type of permutation in which no element appears in its original or designated position. Counting derangements is important when calculating probabilities that involve avoiding fixed points, such as finding the likelihood that no item is matched correctly.

Please give Ace some feedback