Mathematics
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Fluidité · Pack A

11.2 Probability

Répondez à chaque question. Montrez les calculs si nécessaire.

Bronze

  1. A fair die is rolled. Find $P(\text{score is } even)$.

  2. If $P(A) = 0.3$, find $P(A')$.

  3. A bag contains 3 red and 5 blue counters. A counter is drawn at random. Find $P(\text{red})$.

  4. A coin is tossed twice. Find $P(\text{two heads})$.

  5. A spinner is spun 200 times. The result "red" came up 64 times. Estimate $P(\text{ red})$.

  6. A two-way table shows: 25 fish, 35 no fish; 40 meat, 20 no meat. Of 60 respondents, what is $P(\text{meat})$?

  7. Two fair dice are rolled. Find $P(\text{sum} = 7)$.

  8. From a Venn diagram with $n(A) = 12$, $n(B) = 9$, $n(A \cap B) = 4$, $n(U) = 25$, find $P(A)$.

  9. A card is drawn at random from a standard 52-card pack. Find $P(\text{ red card})$.

  10. A spinner has outcomes Red, Blue, Green with $P(R) = 0.4$ and $P(B) = 0.35$. Find $P(G)$.

Silver

  1. Given $P(A) = 0.6$, $P(B) = 0.5$ and $P(A \cap B) = 0.2$, find $P(A \cup B)$.

  2. A bag has 5 red and 4 blue counters. Two are drawn without replacement. Find $P(\text{both red})$.

  3. A bag has 5 red and 4 blue counters. Two are drawn without replacement. Find $P(\text{one of each colour})$.

  4. 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 $P(\text{Spanish})$.

  5. Events $A$ and $B$ are independent with $P(A) = 0.4$ and $P(B) = 0.3$. Find $P(A \cap B)$.

  6. Given $P(A \cap B) = 0.18$ and $P(A) = 0.45$, find $P(B \mid A)$.

  7. A card is drawn from a standard 52-card pack. Find $P(\text{heart or spade})$ and say whether the events are mutually exclusive.

  8. A spinner lands on red with probability 0.4. It is spun 3 times. Find $P(\text{at least one red})$.

  9. Of 100 students, 60 own a phone, 40 own a tablet, 30 own both. A student who owns a tablet is chosen. Find $P(\text{also owns a phone})$.

  10. $A$ and $B$ are independent with $P(A) = 0.5$ and $P(B) = 0.3$. Find $P(A \cup B)$.

Gold

  1. 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 $P(\text{defective})$.

  2. Given $P(A) = 0.4$, $P(B) = 0.25$ and $P(A \cap B) = 0.12$, determine whether $A$ and $B$ are independent.

  3. Three fair coins are tossed. Find $P(\text{exactly one heads})$.

  4. A box has 5 red and 3 blue balls. Three are drawn without replacement. Find $P(\text{all three red})$.

  5. In a survey, $n(A) = 24$, $n(B) = 20$, $n(A \cap B) = 12$ and $n(U) = 50$. Find $P(A \mid B)$.

  6. Given $P(A \cup B) = 0.8$, $P(B) = 0.5$ and $P(A \cap B) = 0.2$, find $P(A)$.

  7. $2\%$ 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 $P(\text{positive test})$.

  8. At a school, $40\%$ study French and $25\%$ study Spanish. Of those who study French, $30\%$ also study Spanish. Find $P(F \cap S)$.

  9. 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 $P(M_1 \mid \text{defective})$.

  10. $A$, $B$, $C$ are mutually independent with $P(A) = P(B) = P(C) = p$. Find $P(\text{exactly one of } A, B, C)$ when $p = \dfrac{1}{3}$.

Platinum

  1. 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 $P(C \mid T^+)$.

  2. In a town, $30\%$ commute by bike; of cyclists $20\%$ are late, of non-cyclists $5\%$ are late. Find $P(\text{cyclist} \mid \text{late})$.

  3. A bag has 4 red, 3 blue and 2 green balls. Two are drawn without replacement. Find $P(\text{second is red} \mid \text{first is not red})$.

  4. A shooter scores with probability $0.6$. How many shots so that $P(\text{at least one score}) \geq 0.99$?

  5. A bag has 3 red, 2 blue and 1 green ball. Two are drawn without replacement. Find $P(\text{same colour})$.

  6. $P(A) = 0.5$, $P(B) = 0.4$ and $P(A \mid B) = 0.6$. Find $P(A \cup B)$ and decide whether $A$ and $B$ are independent.

  7. A vaccine has 70% chance of being effective per person. Find $P(\text{exactly 3 of 5 benefit})$.

  8. A four-digit code is formed using digits $1,2,3,4,5$ without repetition. Find $P(\text{code is even})$.

  9. Two fair dice are rolled. Given that the sum is even, find $P(\text{both dice show the same number})$.

  10. In a group of 50, $n(A) = 30$, $n(B) = 20$ and $n(A \cup B)' = 8$. Find (a) $n(A \cap B)$, (b) $P(A \mid B)$.