Chapter 3: Q.3.9 (page 110)

You ask your neighbor to water a sickly plant while you are on vacation. Without water, it will die with probability . $8$ ; with water, it will die with probability . $15$ . You are $90$ percent certain that your neighbor will remember to water the plant.
(a) What is the probability that the plant will be alive when you return?
(b) If the plant is dead upon your return, what is the probability that your neighbor forgot to water it?

Short Answer

Expert verified

a). The probability that the plant will be alive when you return is $0.785$ .

b). The probability that your neighbor forgot to water it islocalid="1650015942117" $0.3721$ .

Step by step solution

Given Information (Part a)

$W$ -. the plants were watered

$P (D ∣ W^{c}) = 0.8$

$P (D ∣ W) = 0.15$

$P (W) = 0.9$

Explanation (Part a)

Formula of total probability can be applied here (because $W \cap W^{c} = \emptyset, P (W \cup W^{c}) = 1$ ):

$P (D) = P (D ∣ W^{c}) P (W^{c}) + P (D ∣ W) P (W)$

Formula for complement $\Rightarrow P (W^{c}) = 1 - P (W) = 0.1$

Substitution of familiar probabilities:

$P (D) = 0.8 \cdot 0.1 + 0.15 \cdot 0.9 = 0.215$

Now formula for complement applied again:

$P (D^{c}) = 1 - P (D) = 1 - 0.215 = 0.785$

Given Information (Part b)

$W$ -. the plants were watered

$D$ - the plants died

Given probabilities:

P (D ∣ W^{c}) = 0.8

P (D ∣ W) = 0.15

P (W) = 0.9

Explanation (Part b)

The definition of conditional probability gives:

$P (W^{c} ∣ D) = \frac{P (W^{c} \cap D)}{P (D)} and P (D ∣ W^{c}) \cdot P (W^{c}) = P (W^{c} \cap D)$

Which can be fused into:

$P (W^{c} ∣ D) = \frac{P (D ∣ W^{c}) \cdot P (W^{c})}{P (D)}$

In part a) we computed $P (W^{c}) = 0.1, P (D) = 0.215$ is stated in the beginning, therefore:

localid="1650015923077" $P (W^{c} ∣ D) = \frac{0.8 \cdot 0.1}{0.215} = \frac{16}{43} \approx 0.3721$

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Short Answer

Step by step solution

Given Information (Part a)

Explanation (Part a)

Given Information (Part b)

Explanation (Part b)

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Geometry

Applied Mathematics

Decision Maths

Calculus

Discrete Mathematics

Study anywhere. Anytime. Across all devices.