Rose Harrison, Clara Huizink, Aidan Sproat Clements and, Marlene Torres Skoumal Solutions for Exercise 23: Review in context

The probability that a randomly selected person has a bone disorder is $0.01$. The probability that a test for this condition is positive is $0.98$ if the condition is there, and $0.05$ if the condition is not there (a false positive).

Let $B$ be the event 'has the bone disorder' and $T$ be the event 'test is positive'.

Draw a tree diagram to represent these probabilities.

The probability that a randomly selected person has a bone disorder is $0.01$. The probability that a test for this condition is positive is $0.98$. if the condition is there, and $0.05$ if the condition is not there (a false positive).

Let $B$ be the event 'has the bone disorder' and $T$ be the event 'test is positive'.

Calculate the probability of success for the test. Note the test is successful if it tests positive for people with the disorder and negative for people without the disorder.

Michele is making drinks to sell at the school play. She makes a 'Super green smoothie' with baby spinach, cucumber, apple, kiwi and grapes, with a calorific content of $140 \mathrm{kcal}$ per cup, and a 'Triple chocolate milk shake' with chocolate milk, chocolate syrup and chocolate ice-cream, with a calorific content of $370 \mathrm{kcal}$ per cup.

She records the number of $\mathrm{MYP}$ and $\mathrm{DP}$ students buying each product. $\mathrm{MYP}$ students purchase $26$ Super green smoothies and $48$ Triple chocolate milk shakes. $\mathrm{DP}$ students purchase $13$ Super green smoothies and $24$ Triple chocolate milk shakes.

Determine, using both Theorem $2$ and Theorem $3$ , whether the age of the student ($\mathrm{MYP}$ or $\mathrm{DP}$) is independent of their choice of drink. Use a two-way table to answer this question.

A survey of $200000$ people looked at the relationship between cigarette smoking and cancer. Of the $125000$ non-smokers in the survey, $981$ had cancer at some point in their lifetime. Of the smokers in the survey, there were $1763$ people who had cancer.

Find the probability that an individual selected from the study was a smoker who had battled cancer.

A survey of $200000$ people looked at the relationship between cigarette smoking and cancer. Of the $125000$ non-smokers in the survey, $981$ had cancer at some point in their lifetime. Of the smokers in the survey, there were $1763$ people who had cancer.

Find the probability that someone who had never smoked had cancer.

A survey of $200000$ people looked at the relationship between cigarette smoking and cancer. Of the $125000$ non-smokers in the survey, $981$ had cancer at some point in their lifetime. Of the smokers in the survey, there were $1763$ people who had cancer.

Let $B$ be the event 'having cancer' and $A$ be the event 'being a smoker' (includes being a former smoker). Demonstrate whether or not these two events are independent. Justify your answer mathematically.

The following information was extracted from a skin cancer website.

Statement $1$: About $69\%$ of skin cancers are associated with exposure to indoor tanning machines.

Statement $2$: In the United States, $3.3$ million people a year are treated for skin cancer out of a total population of $330$ million.

Statement $3$: $10$ million US adults use indoor tanning machines.

Let $A$ be the event 'developing skin cancer'.

Let $B$ be the event 'exposure to indoor tanning'.

Write down statements $1,2 \mathrm{and} 3$ in probability notation.

The following information was extracted from a skin cancer website.

Statement $1$: About $69\%$ of skin cancers are associated with exposure to indoor tanning machines.

Statement $2$: In the United States, $3.3$ million people a year are treated for skin cancer out of a total population of $330$ million.

Statement $3$: $10$ million US adults use indoor tanning machines.

Let $A$ be the event 'developing skin cancer'.

Let $B$ be the event 'exposure to indoor tanning'.

Determine whether developing skin cancer and exposure to indoor tanning are independent events.