Corrigé
11.2 Probability
Pack A — Réponses
| # | Question | Réponse |
|---|---|---|
| 1 | A fair die is rolled. Find $P(\text{score is } even)$. | $\dfrac{1}{2}$ |
| 2 | If $P(A) = 0.3$, find $P(A')$. | 0.7 |
| 3 | A bag contains 3 red and 5 blue counters. A counter is drawn at random. Find $P(\text{red})$. | $\dfrac{3}{8}$ |
| 4 | A coin is tossed twice. Find $P(\text{two heads})$. | $\dfrac{1}{4}$ |
| 5 | A spinner is spun 200 times. The result "red" came up 64 times. Estimate $P(\text{ red})$. | $\dfrac{8}{25} = 0.32$ |
| 6 | A two-way table shows: 25 fish, 35 no fish; 40 meat, 20 no meat. Of 60 respondents, what is $P(\text{meat})$? | $\dfrac{2}{3}$ |
| 7 | Two fair dice are rolled. Find $P(\text{sum} = 7)$. | $\dfrac{1}{6}$ |
| 8 | From a Venn diagram with $n(A) = 12$, $n(B) = 9$, $n(A \cap B) = 4$, $n(U) = 25$, find $P(A)$. | $\dfrac{12}{25}$ |
| 9 | A card is drawn at random from a standard 52-card pack. Find $P(\text{ red card})$. | $\dfrac{1}{2}$ |
| 10 | A spinner has outcomes Red, Blue, Green with $P(R) = 0.4$ and $P(B) = 0.35$. Find $P(G)$. | 0.25 |
| 11 | Given $P(A) = 0.6$, $P(B) = 0.5$ and $P(A \cap B) = 0.2$, find $P(A \cup B)$. | 0.9 |
| 12 | A bag has 5 red and 4 blue counters. Two are drawn without replacement. Find $P(\text{both red})$. | $\dfrac{5}{18}$ |
| 13 | A bag has 5 red and 4 blue counters. Two are drawn without replacement. Find $P(\text{one of each colour})$. | $\dfrac{5}{9}$ |
| 14 | A two-way table shows: 14 pupils take French only, 8 take Spanish only, 10 take both, 18 take neither. A pupil is chosen at random. Given they take French, find $P(\text{Spanish})$. | $\dfrac{5}{12}$ |
| 15 | Events $A$ and $B$ are independent with $P(A) = 0.4$ and $P(B) = 0.3$. Find $P(A \cap B)$. | 0.12 |
| 16 | Given $P(A \cap B) = 0.18$ and $P(A) = 0.45$, find $P(B \mid A)$. | 0.4 |
| 17 | A card is drawn from a standard 52-card pack. Find $P(\text{heart or spade})$ and say whether the events are mutually exclusive. | $\dfrac{1}{2}$; yes, mutually exclusive |
| 18 | A spinner lands on red with probability 0.4. It is spun 3 times. Find $P(\text{at least one red})$. | 0.784 |
| 19 | Of 100 students, 60 own a phone, 40 own a tablet, 30 own both. A student who owns a tablet is chosen. Find $P(\text{also owns a phone})$. | $\dfrac{3}{4}$ |
| 20 | $A$ and $B$ are independent with $P(A) = 0.5$ and $P(B) = 0.3$. Find $P(A \cup B)$. | 0.65 |
| 21 | A factory has two machines: $M_1$ makes $60\%$ of items with defect rate $2\%$, and $M_2$ makes $40\%$ with defect rate $5\%$. An item is picked at random. Find $P(\text{defective})$. | 0.032 |
| 22 | Given $P(A) = 0.4$, $P(B) = 0.25$ and $P(A \cap B) = 0.12$, determine whether $A$ and $B$ are independent. | Not independent ($P(A)\,P(B) = 0.10 \neq 0.12$) |
| 23 | Three fair coins are tossed. Find $P(\text{exactly one heads})$. | $\dfrac{3}{8}$ |
| 24 | A box has 5 red and 3 blue balls. Three are drawn without replacement. Find $P(\text{all three red})$. | $\dfrac{5}{28}$ |
| 25 | In a survey, $n(A) = 24$, $n(B) = 20$, $n(A \cap B) = 12$ and $n(U) = 50$. Find $P(A \mid B)$. | $\dfrac{3}{5}$ |
| 26 | Given $P(A \cup B) = 0.8$, $P(B) = 0.5$ and $P(A \cap B) = 0.2$, find $P(A)$. | 0.5 |
| 27 | $2\%$ of a population has a condition. A test is positive for $90\%$ of those who have it, and falsely positive for $5\%$ of those who do not. Find $P(\text{positive test})$. | 0.067 |
| 28 | At a school, $40\%$ study French and $25\%$ study Spanish. Of those who study French, $30\%$ also study Spanish. Find $P(F \cap S)$. | 0.12 |
| 29 | Using the factory in G1 (Pack A: $M_1$ 60%/2%, $M_2$ 40%/5%; Pack B: $M_1$ 70%/3%, $M_2$ 30%/6%), find $P(M_1 \mid \text{defective})$. | 0.375 |
| 30 | $A$, $B$, $C$ are mutually independent with $P(A) = P(B) = P(C) = p$. Find $P(\text{exactly one of } A, B, C)$ when $p = \dfrac{1}{3}$. | $\dfrac{4}{9}$ |
| 31 | A diagnostic test: 2% of a population has a condition, $P(T^+ \mid C) = 0.9$, $P(T^+ \mid C') = 0.05$. Given a positive test, find $P(C \mid T^+)$. | $\approx 0.269$ |
| 32 | In a town, $30\%$ commute by bike; of cyclists $20\%$ are late, of non-cyclists $5\%$ are late. Find $P(\text{cyclist} \mid \text{late})$. | $\dfrac{12}{19} \approx 0.632$ |
| 33 | A bag has 4 red, 3 blue and 2 green balls. Two are drawn without replacement. Find $P(\text{second is red} \mid \text{first is not red})$. | $\dfrac{1}{2}$ |
| 34 | A shooter scores with probability $0.6$. How many shots so that $P(\text{at least one score}) \geq 0.99$? | $n = 6$ |
| 35 | A bag has 3 red, 2 blue and 1 green ball. Two are drawn without replacement. Find $P(\text{same colour})$. | $\dfrac{4}{15}$ |
| 36 | $P(A) = 0.5$, $P(B) = 0.4$ and $P(A \mid B) = 0.6$. Find $P(A \cup B)$ and decide whether $A$ and $B$ are independent. | $P(A \cup B) = 0.66$; not independent |
| 37 | A vaccine has 70% chance of being effective per person. Find $P(\text{exactly 3 of 5 benefit})$. | 0.3087 |
| 38 | A four-digit code is formed using digits $1,2,3,4,5$ without repetition. Find $P(\text{code is even})$. | $\dfrac{2}{5}$ |
| 39 | Two fair dice are rolled. Given that the sum is even, find $P(\text{both dice show the same number})$. | $\dfrac{1}{3}$ |
| 40 | In a group of 50, $n(A) = 30$, $n(B) = 20$ and $n(A \cup B)' = 8$. Find (a) $n(A \cap B)$, (b) $P(A \mid B)$. | (a) 8; (b) $\dfrac{2}{5}$ |
Pack B — Réponses
| # | Question | Réponse |
|---|---|---|
| 1 | A fair die is rolled. Find $P(\text{score is } a multiple of 3)$. | $\dfrac{1}{3}$ |
| 2 | If $P(A) = \dfrac{2}{5}$, find $P(A')$. | $\dfrac{3}{5}$ |
| 3 | A bag contains 4 red and 6 blue counters. A counter is drawn at random. Find $P(\text{red})$. | $\dfrac{2}{5}$ |
| 4 | A coin is tossed twice. Find $P(\text{two heads})$. | $\dfrac{1}{2}$ |
| 5 | A spinner is spun 200 times. The result "blue" came up 75 times. Estimate $P(\text{ blue})$. | $\dfrac{3}{8} = 0.375$ |
| 6 | A two-way table shows: 30 fish, 50 no fish; 50 meat, 30 no meat. Of 80 respondents, what is $P(\text{meat})$? | $\dfrac{5}{8}$ |
| 7 | Two fair dice are rolled. Find $P(\text{sum} = 7)$. | $\dfrac{5}{36}$ |
| 8 | From a Venn diagram with $n(A) = 12$, $n(B) = 9$, $n(A \cap B) = 4$, $n(U) = 25$, find $P(A)$. | $\dfrac{4}{25}$ |
| 9 | A card is drawn at random from a standard 52-card pack. Find $P(\text{ face card (J, Q, K)})$. | $\dfrac{3}{13}$ |
| 10 | A spinner has outcomes Red, Blue, Green with $P(R) = 0.4$ and $P(B) = 0.35$. Find $P(G)$. | 0.30 |
| 11 | Given $P(A) = 0.4$, $P(B) = 0.5$ and $P(A \cap B) = 0.15$, find $P(A \cup B)$. | 0.75 |
| 12 | A bag has 6 red and 3 blue counters. Two are drawn without replacement. Find $P(\text{both red})$. | $\dfrac{5}{12}$ |
| 13 | A bag has 4 red and 3 blue counters. Two are drawn without replacement. Find $P(\text{one of each colour})$. | $\dfrac{4}{7}$ |
| 14 | A two-way table shows: 9 pupils take French only, 6 take Spanish only, 6 take both, 9 take neither. A pupil is chosen at random. Given they take French, find $P(\text{Spanish})$. | $\dfrac{2}{5}$ |
| 15 | Events $A$ and $B$ are independent with $P(A) = 0.5$ and $P(B) = 0.6$. Find $P(A \cap B)$. | 0.3 |
| 16 | Given $P(A \cap B) = 0.12$ and $P(A) = 0.4$, find $P(B \mid A)$. | 0.3 |
| 17 | A card is drawn from a standard 52-card pack. Find $P(\text{heart or spade})$ and say whether the events are mutually exclusive. | $\dfrac{3}{4}$; yes, mutually exclusive |
| 18 | A spinner lands on red with probability 0.3. It is spun 4 times. Find $P(\text{at least one red})$. | 0.7599 |
| 19 | Of 100 students, 60 own a phone, 40 own a tablet, 30 own both. A student who owns a tablet is chosen. Find $P(\text{also owns a phone})$. | $\dfrac{2}{3}$ |
| 20 | $A$ and $B$ are independent with $P(A) = 0.5$ and $P(B) = 0.3$. Find $P(A \cup B)$. | 0.55 |
| 21 | A factory has two machines: $M_1$ makes $70\%$ of items with defect rate $3\%$, and $M_2$ makes $30\%$ with defect rate $6\%$. An item is picked at random. Find $P(\text{defective})$. | 0.039 |
| 22 | Given $P(A) = 0.5$, $P(B) = 0.3$ and $P(A \cap B) = 0.15$, determine whether $A$ and $B$ are independent. | Independent ($P(A)\,P(B) = 0.15 = P(A \cap B)$) |
| 23 | Three fair coins are tossed. Find $P(\text{exactly two heads})$. | $\dfrac{3}{8}$ |
| 24 | A box has 6 red and 4 blue balls. Three are drawn without replacement. Find $P(\text{all three red})$. | $\dfrac{1}{6}$ |
| 25 | In a survey, $n(A) = 30$, $n(B) = 25$, $n(A \cap B) = 10$ and $n(U) = 60$. Find $P(A \mid B)$. | $\dfrac{2}{5}$ |
| 26 | Given $P(A \cup B) = 0.7$, $P(B) = 0.4$ and $P(A \cap B) = 0.1$, find $P(A)$. | 0.4 |
| 27 | $3\%$ of a population has a condition. A test is positive for $85\%$ of those who have it, and falsely positive for $4\%$ of those who do not. Find $P(\text{positive test})$. | 0.0643 |
| 28 | At a school, $50\%$ study French and $30\%$ study Spanish. Of those who study French, $20\%$ also study Spanish. Find $P(F \cap S)$. | 0.10 |
| 29 | Using the factory in G1 (Pack A: $M_1$ 60%/2%, $M_2$ 40%/5%; Pack B: $M_1$ 70%/3%, $M_2$ 30%/6%), find $P(M_1 \mid \text{defective})$. | $\dfrac{7}{13} \approx 0.538$ |
| 30 | $A$, $B$, $C$ are mutually independent with $P(A) = P(B) = P(C) = p$. Find $P(\text{exactly one of } A, B, C)$ when $p = \dfrac{1}{4}$. | $\dfrac{27}{64}$ |
| 31 | A diagnostic test: 2% of a population has a condition, $P(T^+ \mid C) = 0.9$, $P(T^+ \mid C') = 0.05$. Given a positive test, find $P(C \mid T^+)$. | $\approx 0.397$ |
| 32 | In a town, $30\%$ commute by bike; of cyclists $20\%$ are late, of non-cyclists $5\%$ are late. Find $P(\text{cyclist} \mid \text{late})$. | $\dfrac{5}{8} = 0.625$ |
| 33 | A bag has 5 red, 2 blue and 3 green balls. Two are drawn without replacement. Find $P(\text{second is red} \mid \text{first is not red})$. | $\dfrac{5}{9}$ |
| 34 | A shooter scores with probability $0.4$. How many shots so that $P(\text{at least one score}) \geq 0.95$? | $n = 6$ |
| 35 | A bag has 3 red, 2 blue and 1 green ball. Two are drawn without replacement. Find $P(\text{same colour})$. | $\dfrac{5}{18}$ |
| 36 | $P(A) = 0.5$, $P(B) = 0.4$ and $P(A \mid B) = 0.6$. Find $P(A \cup B)$ and decide whether $A$ and $B$ are independent. | $P(A \cup B) = 0.9$; not independent |
| 37 | A vaccine has 70% chance of being effective per person. Find $P(\text{exactly 3 of 5 benefit})$. | 0.3456 |
| 38 | A four-digit code is formed using digits $1,2,3,4,5$ without repetition. Find $P(\text{code is even})$. | $\dfrac{1}{20}$ |
| 39 | Two fair dice are rolled. Given that the sum is even, find $P(\text{both dice show the same number})$. | $\dfrac{1}{3}$ |
| 40 | In a group of 50, $n(A) = 30$, $n(B) = 20$ and $n(A \cup B)' = 8$. Find (a) $n(A \cap B)$, (b) $P(A \mid B)$. | (a) 10; (b) $\dfrac{2}{5}$ |
Problèmes — Solutions détaillées
**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.
(a) $P(A) = \frac{1}{2}$, $P(B) = \frac{1}{2}$, $P(A \cap B) = \frac{1}{3}$, $P(A \cup B) = \frac{2}{3}$. (b) Not independent: $P(A)P(B) = \frac{1}{4} \neq \frac{1}{3}$.
**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 $P(\text{exactly one red})$.
(a) Each branch: $P(R) = \frac{4}{7}$, $P(B) = \frac{3}{7}$. (b) $\frac{16}{49}$. (c) $\frac{24}{49}$.
**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 $P(\text{one of each colour})$.
(a) Branches scale on the second pick. (b) $\frac{1}{6}$. (c) $\frac{5}{9}$.
**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.
(a) See working. (b) 0.067. (c) $P(C \mid T^+) \approx 0.269$.
**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.
(a) 0.12. (b) 0.53. (c) Not independent.
**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.
(a) 0.032. (b) 0.625. (c) Wrong — 62.5% of defectives come from $M_2$.
**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 $P(\text{exactly 3 benefit})$.
(a) 0.343. (b) $n = 6$. (c) 0.3087.
**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 $P(\text{code starts with 1 and ends with 5})$.
(a) 120. (b) $\frac{2}{5}$. (c) $\frac{1}{20}$.
**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.
(a) Tea only 18; both 12; coffee only 13; neither 7. (b) $\frac{30}{50}, \frac{25}{50}, \frac{12}{50}, \frac{7}{50}$. (c) $P(T \mid C) = \frac{12}{25}$. (d) Not independent.
**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?
(a) 10%. (b) $\frac{1}{5}$. (c) Not independent. (d) Both 28%, neither 18%.
**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 $P(\text{phone})$.
(a) $\frac{3}{4}, \frac{2}{5}, \frac{3}{10}$. (b) Yes: $\frac{3}{4} \cdot \frac{2}{5} = \frac{3}{10}$ ✓. (c) $\frac{3}{4}$.
**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.
(a) 0.022. (b) $\frac{15}{22} \approx 0.682$. (c) Wrong — Y's defect rate is 5× X's, so 68% of defectives come from Y.