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Chapter 3: Problem 90

Consider the following scenario: Let P(C) = 0.4. Let P(D) = 0.5. Let P(C|D) = 0.6. a. Find P(C AND D). b. Are C and D mutually exclusive? Why or why not? c. Are C and D independent events? Why or why not? d. Find P(C OR D). e. Find P(D|C)

Short Answer

Expert verified
a. 0.3; b. No; c. No; d. 0.6; e. 0.75.

Step by step solution

01

Calculate P(C AND D)

To find the probability of both C and D occurring, use the formula for conditional probability: \( P(C \text{ AND } D) = P(D) \times P(C|D) \). Substitute the known values: \( P(D) = 0.5 \) and \( P(C|D) = 0.6 \). Therefore, \( P(C \text{ AND } D) = 0.5 \times 0.6 = 0.3 \).
02

Check if C and D are Mutually Exclusive

Events are mutually exclusive if they cannot occur at the same time, which means \( P(C \text{ AND } D) = 0 \). From Step 1, we have \( P(C \text{ AND } D) = 0.3 \). Since this is not zero, C and D are not mutually exclusive.
03

Check if C and D are Independent Events

Events are independent if \( P(C \text{ AND } D) = P(C) \times P(D) \). We have \( P(C \text{ AND } D) = 0.3 \) and need to check it against \( P(C) \times P(D) = 0.4 \times 0.5 = 0.2 \). Since \( 0.3 eq 0.2 \), C and D are not independent.
04

Calculate P(C OR D)

Use the formula for the probability of either event occurring: \( P(C \text{ OR } D) = P(C) + P(D) - P(C \text{ AND } D) \). Substitute the known probabilities: \( P(C) = 0.4 \), \( P(D) = 0.5 \), and \( P(C \text{ AND } D) = 0.3 \). Thus, \( P(C \text{ OR } D) = 0.4 + 0.5 - 0.3 = 0.6 \).
05

Calculate P(D|C)

To find \( P(D|C) \), use the formula for conditional probability: \( P(D|C) = \frac{P(C \text{ AND } D)}{P(C)} \). With \( P(C \text{ AND } D) = 0.3 \) and \( P(C) = 0.4 \), we find \( P(D|C) = \frac{0.3}{0.4} = 0.75 \).

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Key Concepts

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

Mutually Exclusive Events
In probability, mutually exclusive events are those that cannot happen at the same time. Think about tossing a coin 鈥� you can't get both heads and tails on the same flip. If events C and D were mutually exclusive, the probability of both occurring together, denoted as \( P(C \text{ AND } D) \), would be zero.In our scenario, \( P(C \text{ AND } D) \) was found to be 0.3. This is a clear indicator that C and D are not mutually exclusive because both can occur at the same time. Understanding mutually exclusive events helps in probability calculations because if two events are mutually exclusive, the probability of either one occurring is simply the sum of their individual probabilities. However, we can't apply this concept when events are not mutually exclusive like C and D in our example.
Independent Events
Independent events are a key concept in probability. These are events where the occurrence of one does not affect the probability of the other occurring. If events C and D were independent, the probability that both C and D occur, which is \( P(C \text{ AND } D) \), would be equal to the product of their individual probabilities, \( P(C) \times P(D) \).In this instance, we calculated \( P(C \text{ AND } D) = 0.3 \) and compared it with \( P(C) \times P(D) = 0.4 \times 0.5 = 0.2 \). Since 0.3 is not equal to 0.2, events C and D are not independent.Recognizing independent events is crucial when dealing with probability calculations because it simplifies the problem-solving process. When events are independent, you can treat each event as if the others don鈥檛 exist, leading to simpler calculations.
Probability Calculations
Probability calculations involve determining the likelihood of various outcomes. For our scenario, we were tasked with finding different probabilities involving events C and D. Each type of probability requires a specific approach and formula.- **Calculating Probability of Both Events:** To find \( P(C \text{ AND } D) \), the conditional probability formula was used: \( P(C \text{ AND } D) = P(D) \times P(C|D) \). By substituting the given values, we calculated this probability as 0.3.- **Probability of Either Event Occurring:** This involves \( P(C \text{ OR } D) \). We used the inclusion-exclusion principle: \( P(C \text{ OR } D) = P(C) + P(D) - P(C \text{ AND } D) \). For C and D, it resulted in 0.6.To make accurate probability calculations, it's essential to know which formulas apply to the given conditions, such as whether events are mutually exclusive or independent. This structured approach helps in accurately determining the likelihood of scenarios.

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Most popular questions from this chapter

The following table of data obtained from www.baseball-almanac.com shows hit information for four players. Suppose that one hit from the table is randomly selected. $$\begin{array}{|l|l|l|l|l|}\hline \text { Name } & {\text { single }} & {\text { Double }} & {\text { Triple }} & {\text { Home Run }} & {\text { Total Hits }} \\ \hline \text { Babe Ruth } & {1,517} & {506} & {136} & {714} & {2,873} \\ \hline \text { Jackie Robinson } & {1,054} & {273} & {54} & {137} & {1,518} \\ \hline \text { Ty Cobb } & {3,603} & {174} & {295} & {114} & {4,189} \\ \hline \text { Hank Aaron } & {2,294} & {624} & {98} & {755} & {3,771} \\ \hline\end{array}$$ Are "the hit being made by Hank Aaron" and "the hit being a double" independent events? a. Yes, because P(hit by Hank Aaron|hit is a double) = P(hit by Hank Aaron) b. No, because P(hit by Hank Aaron|hit is a double) ? P(hit is a double) c. No, because P(hit is by Hank Aaron|hit is a double) ? P(hit by Hank Aaron) d. Yes, because P(hit is by Hank Aaron|hit is a double) = P(hit is a double)

Use the following information to answer the next six exercises. A jar of 150 jelly beans contains 22 red jelly beans, 38 yellow, 20 green, 28 purple, 26 blue, and the rest are orange. Find \(P(Y)\)

Use the following information to answer the next six exercises. There are 23 countries in North America, 12 countries in South America, 47 countries in Europe, 44 countries in Asia, 54 countries in Africa, and 14 in Oceania (Pacific Ocean region). Let A = the event that a country is in Asia. Let E = the event that a country is in Europe. Let F = the event that a country is in Africa. Let N = the event that a country is in North America. Let O = the event that a country is in Oceania. Let S = the event that a country is in South America. Find P(N).

1994, the U.S. government held a lottery to issue 55,000 Green Cards (permits for non-citizens to work legally in the U.S.). Renate Deutsch, from Germany, was one of approximately 6.5 million people who entered this lottery. Let G = won green card. a. What was Renate鈥檚 chance of winning a Green Card? Write your answer as a probability statement. b. In the summer of 1994, Renate received a letter stating she was one of 110,000 finalists chosen. Once the finalists were chosen, assuming that each finalist had an equal chance to win, what was Renate鈥檚 chance of winning a Green Card? Write your answer as a conditional probability statement. Let F = was a finalist. c. Are G and F independent or dependent events? Justify your answer numerically and also explain why. d. Are G and F mutually exclusive events? Justify your answer numerically and explain why.

On February 28, 2013, a Field Poll Survey reported that 61% of California registered voters approved of allowing two people of the same gender to marry and have regular marriage laws apply to them. Among 18 to 39 year olds (California registered voters), the approval rating was 78%. Six in ten California registered voters said that the upcoming Supreme Court鈥檚 ruling about the constitutionality of California鈥檚 Proposition 8 was either very or somewhat important to them. Out of those CA registered voters who support same-sex marriage, 75% say the ruling is important to them. In this problem, let: 鈥� C = California registered voters who support same-sex marriage. 鈥� B = California registered voters who say the Supreme Court鈥檚 ruling about the constitutionality of California鈥檚 Proposition 8 is very or somewhat important to them 鈥� A = California registered voters who are 18 to 39 years old. a. Find P(C). b. Find P(B). c. Find P(C|A). d. Find P(B|C). e. In words, what is C|A? f. In words, what is B|C? g. Find P(C AND B). h. In words, what is C AND B? i. Find P(C OR B). j. Are C and B mutually exclusive events? Show why or why not.
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