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Chapter 5: Problem 32

Current estimates are that about \(25 \%\) of all deaths are due to cancer, and of the deaths that are due to cancer, \(30 \%\) are attributed to tobacco, \(40 \%\) to diet, and \(30 \%\) to other causes. a. Define events, and identify which of these four probabilities refer to conditional probabilities. b. Find the probability that a death is due to cancer and tobacco.

Short Answer

Expert verified
The probability that a death is due to cancer and tobacco is 0.075.

Step by step solution

01

Define Events

Let's start by defining the events. Let event \( C \) be a death due to cancer. Event \( T \) is a death due to tobacco, \( D \) is a death due to diet, and \( O \) is a death due to other causes. The given probabilities are \( P(C) = 0.25 \), \( P(T|C) = 0.30 \), \( P(D|C) = 0.40 \), and \( P(O|C) = 0.30 \).
02

Identify Conditional Probabilities

The conditional probabilities are \( P(T | C) = 0.30 \), \( P(D | C) = 0.40 \), and \( P(O | C) = 0.30 \). These probabilities are dependent on the event that a death is due to cancer and represent the breakdown of causes attributed to those cancer deaths.
03

Calculate Probability of Death Due to Cancer and Tobacco

To find the probability that a death is due to cancer and tobacco, use the formula for conditional probabilities: \( P(T \cap C) = P(T|C) \times P(C) \). Substitute the known values: \( P(T \cap C) = 0.30 \times 0.25 = 0.075 \).

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Key Concepts

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

Event Definition
In probability theory, the concept of an "event" is a fundamental idea. An event is any outcome or set of outcomes of a random experiment. Let's consider the context of our exercise, which involves deaths due to various causes. Here, we define specific events:
  • Event \( C \): A death due to cancer.
  • Event \( T \): A death due to tobacco.
  • Event \( D \): A death due to diet.
  • Event \( O \): A death due to other causes.
These definitions are crucial because they help us categorize and calculate probabilities related to each specific cause, allowing us to conduct detailed analysis. By clearly defining these events, we can apply mathematical formulas and understand the relationships between different occurrences in our probability study.
Probability Calculation
Probability calculation allows us to quantify the likelihood of events occurring. In our exercise, we are provided with several probabilities. The probability of a death due to cancer is given as \( P(C) = 0.25 \). This means there's a 25% chance that any given death is due to cancer.Next, within the context of cancer-related deaths, we are given breakdown probabilities:
  • \( P(T|C) = 0.30 \): The probability of a death due to tobacco given it is already due to cancer.
  • \( P(D|C) = 0.40 \): The probability of a death due to diet given it is already due to cancer.
  • \( P(O|C) = 0.30 \): The probability of a death due to other causes given it is already due to cancer.
These breakdowns help us understand the distribution of specific causes within the broader category of cancer deaths. Calculating these probabilities involves multiplying frequencies and considering conditional probabilities related to specific events.
Conditional Probability Formula
Conditional probability allows us to calculate the probability of an event occurring given that another event has already occurred. The conditional probability formula is expressed as:\[ P(A|B) = \frac{P(A \cap B)}{P(B)} \]where:
  • \( P(A|B) \) is the conditional probability of event \( A \) occurring given that \( B \) has occurred.
  • \( P(A \cap B) \) is the probability of both events \( A \) and \( B \) occurring.
  • \( P(B) \) is the probability of event \( B \) occurring.
In our exercise, we wanted to find \( P(T \cap C) \), the probability of a death due to both cancer and tobacco. Using the formula: \[ P(T \cap C) = P(T|C) \times P(C) \]we substitute the known values to obtain:\[ P(T \cap C) = 0.30 \times 0.25 = 0.075 \]This result tells us that there's a 7.5% chance a death is due to both cancer and tobacco 鈥� providing insight into how these events overlap.

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Most popular questions from this chapter

A teacher announces a pop quiz for which the student is completely unprepared. The quiz consists of 100 true-false questions. The student has no choice but to guess the answer randomly for all 100 questions. a. Simulate taking this quiz by random guessing. Number a sheet of paper 1 to 100 to represent the 100 questions. Write a \(\mathrm{T}\) (true) or \(\mathrm{F}\) (false) for each question, by predicting what you think would happen if you repeatedly flipped a coin and let a tail represent a T guess and a head represent an F guess. (Don't actually flip a coin, but merely write down what you think a random series of guesses would look like.) b. How many questions would you expect to answer correctly simply by guessing? c. The table shows the 100 correct answers. The answers should be read across rows. How many questions did you answer correctly? d. The above answers were actually randomly generated by the Simulating the Probability of Head With a Fair Coin applet on the text CD. What percentage were true, and what percentage would you expect? Why are they not necessarily identical? e. Are there groups of answers within the sequence of 100 answers that appear nonrandom? For instance, what is the longest run of Ts or Fs? By comparison, which is the longest run of Ts or Fs within your sequence of 100 answers? (There is a tendency in guessing what randomness looks like to identify too few long runs in which the same outcome occurs several times in a row.)

Larry Bird, who played pro basketball for the Boston Celtics, was known for being a good shooter. In games during \(1980-1982,\) when he missed his first free throw, 48 out of 53 times he made the second one, and when he made his first free throw, 251 out of 285 times he made the second one. a. Form a contingency table that cross tabulates the outcome of the first free throw (made or missed) in the rows and the outcome of the second free throw (made or missed) in the columns. b. For a given pair of free throws, estimate the probability that Bird (i) made the first free throw and (ii) made the second free throw. c. Estimate the probability that Bird made the second free throw, given that he made the first one. Does it seem as if his success on the second shot depends strongly, or hardly at all, on whether he made the first?

A standard card deck has 52 cards consisting of 26 black and 26 red cards. Three cards are dealt from a shuffled deck, without replacement. a. True or false: The probability of being dealt three black cards is \((1 / 2) \times(1 / 2) \times(1 / 2)=1 / 8 .\) If true, explain why. If false, show how to get the correct probability. b. Let \(A=\) first card red and \(B=\) second card red. Are A and B independent? Explain why or why not. c. Answer parts a and b if each card is replaced in the deck after being dealt.

A couple plans on having three children. Suppose that the probability of any given child being female is \(0.5,\) and also suppose that the genders of each child are independent events. a. Write out all outcomes in the sample space for the genders of the three children. b. What should be the probability associated with each outcome? Using the sample space constructed in part a, find the probability that the couple will have c. two girls and one boy. d. at least one child of each gender.

State an event that has happened to you or to someone you know that seems highly coincidental (such as seeing a friend while on vacation). Explain why that event may not be especially surprising, once you think of all the similar types of events that could have happened to you or someone that you know, over the course of several years.
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