91Ó°ÊÓ

About \(26 \%\) of movies coming out of Hollywood are comedies, Warner Bros has been the lead studio for about \(13 \%\) of recent movies, and about \(3 \%\) of recent movies are comedies from Warner Bros. \(^{2}\) Let \(\mathrm{C}\) denote the event a movie is a comedy and \(W\) denote the event a movie is produced by Warner Bros. (a) Write probability expressions for each of the three facts given in the first sentence of the exercise. (b) What is the probability that a movie is either a comedy or produced by Warner Bros? (c) What is the probability that a Warner Bros movie is a comedy? (d) What is the probability that a comedy has Warner Bros as its producer? (e) What is the probability that a movie coming out of Hollywood is not a comedy? (f) In terms of movies, what would it mean to say that \(\mathrm{C}\) and \(\mathrm{W}\) are disjoint events? Are they disjoint events? (g) In terms of movies, what would it mean to say that \(\mathrm{C}\) and \(\mathrm{W}\) are independent events? Are they independent events?

Step by step solution

01

Understanding the problem

Let C denote the event a movie is a comedy, W denote the event a movie is produced by Warner Bros. The following are given: P(C) = 0.26, P(W) = 0.13, and P(C ∩ W) = 0.03. (a), (b), (c), (d), and (e) seek to identify probabilities of several events while (f) and (g) analyze the relationship between the two events mentioned, C (comedies) and W (produced by Warner Bros).

02

Calculation of probability expressions

(a) P(C) = 26/100, P(W) = 13/100, and P(C ∩ W) = 3/100.

03

Probability of movie being a comedy or produced by Warner Bros

(b) P(C ∪ W) = P(C) + P(W) - P(C ∩ W) = 0.26 + 0.13 - 0.03 = 0.36.

04

Probability that a Warner Bros movie is a comedy

(c) P(C | W) = P(C ∩ W) / P(W) = 0.03/0.13 ≈ 0.231.

05

Probability that a comedy has Warner Bros as its producer

(d) P(W | C) = P(C ∩ W) / P(C) = 0.03/0.26 ≈ 0.115.

06

Probability that a movie is not a comedy

(e) P(not C) = 1 - P(C) = 1 - 0.26 = 0.74.

07

Interpreting disjoint events

(f) If C and W are disjoint events, it would mean that no comedy is produced by Warner Bros. They are not disjoint, since P(C ∩ W) = 0.03 > 0.

08

Interpreting independent events

(g) If C and W are independent events, it would mean that the outcome of one does not affect the outcome of the other. They are not independent, since P(C ∩ W) ≠ P(C)P(W).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Disjoint Events

In probability, disjoint events, also known as mutually exclusive events, are events that cannot happen at the same time. For instance, when you roll a single dice, getting a 3 and a 5 at the same time on a single roll is impossible, hence they are disjoint.

In terms of movies, if event C represents a movie being a comedy and event W represents a movie being produced by Warner Bros, the events would be disjoint if no Warner Bros movie could be a comedy. This means the intersection of the two events, denoted by \(P(C \cap W)\), would be zero.

However, based on our given data, \(P(C \cap W) = 0.03\), which means 3% of movies are comedies produced by Warner Bros. Therefore, C and W are not disjoint events. Simply put, a movie can be both a comedy and be produced by Warner Bros, so these events do indeed overlap.

• Disjoint events cannot occur simultaneously.
• C and W are not disjoint as they do occur together for some cases.

Independent Events

Independent events in probability are those where the occurrence of one event does not affect the occurrence of another. A classic example is flipping a coin; getting heads on one flip does not change the odds of getting heads on the next flip.

For the movie example, if the events C (a movie being a comedy) and W (a movie produced by Warner Bros) were independent, the probability of their intersection \(P(C \cap W)\) would equal the product of their individual probabilities, expressed as \(P(C) \times P(W)\).

Given our values, \(P(C \cap W) = 0.03\) but \(P(C) \times P(W) = 0.26 \times 0.13 = 0.0338\). Since \(0.03 eq 0.0338\), these events are not independent. This indicates that having a movie be a comedy does influence whether it's produced by Warner Bros or vice versa, at least to some extent.

• Independent events have no impact on each other.
• C and W are not independent, as shown by their unequal probability relationship.

Conditional Probability

Conditional probability reflects the likelihood of one event given that another event has already occurred. It is symbolized as \(P(A|B)\), meaning the probability of event A occurring given event B has occurred.

For our Hollywood example, let's dive into two situations:

1. **Probability of a Warner Bros movie being a comedy**
This is given as \(P(C|W) = \frac{P(C \cap W)}{P(W)}\). Hence, with \(P(C \cap W) = 0.03\) and \(P(W) = 0.13\), it follows that \(P(C|W) = \frac{0.03}{0.13} \approx 0.231\).

2. **Probability of a comedy being produced by Warner Bros**
Here, \(P(W|C) = \frac{P(C \cap W)}{P(C)}\), leading us to \(P(W|C) = \frac{0.03}{0.26} \approx 0.115\).

These calculations show how knowing one attribute (like a movie being a Warner Bros production) affects the probability of another attribute (such as the movie being a comedy).

• Conditional probability considers the relationship between two events.
• Knowing one event can alter the perceived likelihood of another.