Question

Consider the following information about travelers on vacation: \(40 \%\) check work email, \(30 \%\) use a cell phone to stay connected to work, \(25 \%\) bring a laptop with them on vacation, \(23 \%\) both check work email and use a cell phone to stay connected, and \(51 \%\) neither check work email nor use a cell phone to stay connected nor bring a laptop. In addition \(88 \%\) of those who bring a laptop also check work email and \(70 \%\) of those who use a cell phone to stay connected also bring a laptop. With \(E=\) event that a traveler on vacation checks work email, \(C=\) event that a traveler on vacation uses a cell phone to stay connected, and \(L=\) event that a traveler on vacation brought a laptop, use the given information to determine the following probabilities. A Venn diagram may help. a. \(P(E)\) b. \(P(C)\) c. \(P(L)\) d. \(P(E\) and \(C)\) e. \(P\left(E^{C}\right.\) and \(C^{C}\) and \(L^{C}\) ) f. \(P(\) Eor C or \(L\) ) g. \(P(E \mid L)\) j. \(P(E\) and \(L)\) h. \(P(L \mid C)\) k. \(P(C\) and \(L)\) i. \(P(E\) and \(C\) and \(L)\) 1\. \(P(C \mid E\) and \(L)\)

Short answer

a. 0.40, b. 0.30, c. 0.25, d. 0.23, e. 0.51, f. 0.49, g. 0.88, h. 0.70, i. 0.18, 1. 0.82

Step-by-step solution

Identify the basic probabilities

Start by identifying the probabilities given directly in the question. a. \(P(E) = 0.40\) - The probability that a traveler checks their work email. b. \(P(C) = 0.30\) - The probability that a traveler uses a cell phone to stay connected to work.c. \(P(L) = 0.25\) - The probability a traveler brings a laptop on vacation.

Identify the intersecting probabilities

Next, identify the group probabilities i.e. the probabilities that combine more than one condition.d. \(P(E \cap C) = 0.23\) - The probability that a traveler checks their work email and uses a cell phone to stay connected.j. \(P(E \cap L) = 0.88 \cdot P(L) = 0.22\) - The probability that a traveler checks their work email and also brings a laptop.k. \(P(C \cap L) = 0.70 \cdot P(C) = 0.21\) - The probability that a traveler uses a cell phone and also brings a laptop.

Identify the complementary probabilities

Identify the complementary probabilities, i.e., the probabilities of the events not happening.e. \(P(E^{C} \cap C^{C} \cap L^{C}) = 0.51\) - The probability that a traveler neither checks their work email nor uses a cell phone to stay connected nor brings a laptop.

Identify the combined probabilities

Next, identify the combined probabilities, i.e., the probabilities of any of the events happening.f. \(P(E \cup C \cup L) = 1 - P(E^{C} \cap C^{C} \cap L^{C}) = 1 - 0.51 = 0.49\)

Identify the conditional probabilities

Next, identify the conditional probabilities, i.e., the probabilities that depend on the prior occurrence of another event.g. \(P(E|L) = \frac{P(E \cap L)}{P(L)} = \frac{0.22}{0.25} = 0.88\) - The probability that a traveler checks their work email given that they bring a laptop.h. \(P(L|C) = \frac{P(L \cap C)}{P(C)} = \frac{0.21}{0.30} = 0.70\) - The probability that a traveler brings a laptop given that they use a cell phone to stay connected.i. \(P(E \cap C \cap L) = P(E|L) \cdot P(L|C) \cdot P(C) = 0.88 \cdot 0.70 \cdot 0.30 = 0.18\) - The probability that a traveler checks their work email, uses a cell phone, and brings a laptop.1. \(P(C|E \cap L) = \frac{P(E \cap C \cap L)}{P(E \cap L)} = \frac{0.18}{0.22} = 0.82\) - The probability that a traveler uses a cell phone to stay connected given that they checks work email and brings a laptop.