A large survey indicates that 7% of the US adult population uses some form of tobacco product. Researchers have shown that 27% of US adult tobacco users develop lung cancer. Comparatively, only 4% of US adults who do not use tobacco products develop lung cancer.

a) If a US adult is selected at random, what is the probability that the adult uses tobacco products and has lung cancer? Give your answer to four decimal places. 


b) If a US adult is selected at random, what is the probability that the adult has lung cancer? Give your answer to four decimal places. 


c) If a US adult is selected at random, what is the probability that the adult uses tobacco products given that they have lung cancer? Give your answer to four decimal places.

We can solve these problems using basic probability rules and the given information:

### Information Given:
- 7% of the U.S. adult population uses some form of tobacco product. This is the probability that an adult uses tobacco, denoted as \( P(T) = 0.07 \).
- 27% of U.S. adult tobacco users develop lung cancer. This is the probability of having lung cancer given that the person uses tobacco, denoted as \( P(C \mid T) = 0.27 \).
- 4% of U.S. adults who do not use tobacco products develop lung cancer. This is the probability of having lung cancer given that the person does not use tobacco, denoted as \( P(C \mid \overline{T}) = 0.04 \).
- The probability of not using tobacco is \( P(\overline{T}) = 1 - 0.07 = 0.93 \).

### a) Probability that the adult uses tobacco products **and** has lung cancer:

We need to find the probability that an adult both uses tobacco and has lung cancer, denoted as \( P(T \cap C) \). This is given by the multiplication rule:

\[
P(T \cap C) = P(T) \cdot P(C \mid T)
\]

Substitute the values:

\[
P(T \cap C) = 0.07 \cdot 0.27 = 0.0189
\]

So, the probability that the adult uses tobacco products and has lung cancer is:

\[
\boxed{0.0189}
\]

### b) Probability that the adult has lung cancer:

We use the law of total probability to find the overall probability of having lung cancer, denoted as \( P(C) \). This can be calculated as the sum of the probabilities of having lung cancer for tobacco users and non-users:

\[
P(C) = P(T) \cdot P(C \mid T) + P(\overline{T}) \cdot P(C \mid \overline{T})
\]

Substitute the values:

\[
P(C) = (0.07 \cdot 0.27) + (0.93 \cdot 0.04)
\]
\[
P(C) = 0.0189 + 0.0372 = 0.0561
\]

So, the probability that the adult has lung cancer is:

\[
\boxed{0.0561}
\]

### c) Probability that the adult uses tobacco given that they have lung cancer:

We need to find the conditional probability that an adult uses tobacco given that they have lung cancer, denoted as \( P(T \mid C) \). This is given by Bayes' Theorem:

\[
P(T \mid C) = \frac{P(T \cap C)}{P(C)}
\]

We already know \( P(T \cap C) = 0.0189 \) and \( P(C) = 0.0561 \), so:

\[
P(T \mid C) = \frac{0.0189}{0.0561} = 0.3368
\]

So, the probability that the adult uses tobacco given that they have lung cancer is:

\[
\boxed{0.3368}
\]

---

Do you want further details or have any questions?

Here are 5 related questions to deepen understanding:
1. How would the results change if the percentage of tobacco users increased to 10%?
2. What is the probability that a non-tobacco user develops lung cancer?
3. How can these probabilities be used to estimate public health risks?
4. If 30% of lung cancer patients quit smoking, how would that affect the overall probabilities?
5. How could this model be adapted for a population with different smoking habits?

**Tip:** Conditional probability, like in part (c), is very useful in medical diagnostics for estimating risks based on prior information.

A large survey indicates that 7% of the US adult population uses some form of tobacco product. Researchers have shown that 27% of US adult tobacco users develop lung cancer. Comparatively, only 4% of US adults who do not use tobacco products develop lung cancer.

 a) If a US adult is selected at random, what is the probability that the adult uses tobacco products and has lung cancer? Give your answer to four decimal places. 

b) If a US adult is selected at random, what is the probability that the adult has lung cancer? Give your answer to four decimal places. 

c) If a US adult is selected at random, what is the probability that the adult uses tobacco products given that they have lung cancer? Give your answer to four decimal places.

Explore how to calculate the probability of lung cancer among tobacco users and non-users using conditional probability and the law of total probability. Learn step-by-step solutions using Bayes' Theorem. Suitable for students in Grades 10-12.

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