11.2 Probability

**Die events.** A fair die is rolled. Let $A$ = "score is even" and $B$ = "score is greater than 3". (a) Find $P(A)$, $P(B)$, $P(A \cap B)$ and $P(A \cup B)$. (b) Are $A$ and $B$ independent? Justify.

**Tree with replacement.** A bag contains 4 red and 3 blue counters. Two are drawn one at a time, **with** replacement. (a) Draw a tree diagram. (b) Find $P(\text{both red})$. (c) Find $P(\text{exactly one red})$.

**Without replacement.** A bag contains 5 red and 4 blue marbles. Two are drawn **without** replacement. (a) Draw a tree diagram showing all four paths. (b) Find $P(\text{both blue})$. (c) Find $P(\text{one of each colour})$.

**Diagnostic test (rare condition).** A test is used to detect a rare condition. - 2% of people have the condition. - 90% of those who have it test positive. - 5% of those who do not have it test positive (false positives). (a) Draw a tree diagram with the four outcomes. (b) Find $P(\text{positive test})$. (c) Given a positive test, find $P(\text{has the condition})$. Comment briefly.

**Languages — independence.** At a school, 40% of students study French ($F$) and 25% study Spanish ($S$). Of those who study French, 30% also study Spanish. (a) Find $P(F \cap S)$. (b) Find $P(F \cup S)$. (c) Are $F$ and $S$ independent? Justify.

**Reverse conditional.** Two machines, $M_1$ and $M_2$, produce identical items. $M_1$ makes 60% of items with defect rate 2%; $M_2$ makes 40% with defect rate 5%. (a) Find $P(\text{defective})$. (b) Given a defective item, find $P(M_2 \mid \text{defective})$. (c) A buyer claims "most defectives come from $M_1$ because $M_1$ makes more items." Critique this claim.

**At least one — design question.** A vaccine has a 70% chance of being effective per person, treated as independent trials. (a) Find the probability that the first 3 people all benefit. (b) Find the smallest $n$ such that $P(\text{at least one benefits}) \geq 0.999$. (c) If 5 people are vaccinated, find $P(\text{exactly 3 benefit})$.

**Combinatorial code.** A four-digit code is formed using the digits $1,2,3,4,5$ **without** repetition. (a) How many different codes are possible? (b) Find $P(\text{code is even})$. (c) Find $P(\text{code starts with 1 and ends with 5})$.

**Conditional on a Venn.** In a survey of 50 people, 30 like tea, 25 like coffee, and 12 like both. (a) Construct a 2-set Venn diagram and fill in all four regions. (b) Find $P(\text{tea}) $, $P(\text{coffee})$, $P(\text{both})$ and $P(\text{neither})$. (c) Given the person likes coffee, find $P(\text{also tea})$. (d) Are "likes tea" and "likes coffee" independent? Justify.

**Mixed scenario — sport, music & both.** At a youth club, every member plays sport ($S$), studies music ($M$), or both. 60% play sport, 50% study music. (a) Show that 10% do both. (b) Find $P(\text{plays sport} \mid \text{studies music})$. (c) Are sport and music independent at this club? Justify. (d) A different club has 70% sport and 40% music, genuinely independent. What percentage does both, and what percentage neither?

**Two-way table — phone survey.** A sample of 200 students reports phone ownership and tablet ownership. | | Tablet | No tablet | Total | |--------------|--------|-----------|-------| | Phone | 60 | 90 | 150 | | No phone | 20 | 30 | 50 | | Total | 80 | 120 | 200 | (a) Find $P(\text{phone})$, $P(\text{tablet})$, $P(\text{phone and tablet})$. (b) Are "phone" and "tablet" independent? Justify with a calculation. (c) Given a student owns a tablet, find $P(\text{phone})$.

**Reverse conditional in context.** A factory's QA pipeline classifies items as defective (D) or fine. Two product lines (X and Y) feed the same conveyor: - 70% of items come from line X with defect rate 1%. - 30% of items come from line Y with defect rate 5%. (a) An item is taken at random. Find $P(D)$. (b) Given the item is defective, find $P(Y \mid D)$. (c) Quality control says "since most output is from X, most defectives are from X." Critique using your answers.