11.2 Probability

A fair die is rolled. Find $P(\text{score is } a multiple of 3)$.

If $P(A) = \dfrac{2}{5}$, find $P(A')$.

A bag contains 4 red and 6 blue counters. A counter is drawn at random. Find $P(\text{red})$.

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

A spinner is spun 200 times. The result "blue" came up 75 times. Estimate $P(\text{ blue})$.

A two-way table shows: 30 fish, 50 no fish; 50 meat, 30 no meat. Of 80 respondents, what is $P(\text{meat})$?

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

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

A card is drawn at random from a standard 52-card pack. Find $P(\text{ face card (J, Q, K)})$.

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

Given $P(A) = 0.4$, $P(B) = 0.5$ and $P(A \cap B) = 0.15$, find $P(A \cup B)$.

A bag has 6 red and 3 blue counters. Two are drawn without replacement. Find $P(\text{both red})$.

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

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 $P(\text{Spanish})$.

Events $A$ and $B$ are independent with $P(A) = 0.5$ and $P(B) = 0.6$. Find $P(A \cap B)$.

Given $P(A \cap B) = 0.12$ and $P(A) = 0.4$, find $P(B \mid A)$.

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

A spinner lands on red with probability 0.3. It is spun 4 times. Find $P(\text{at least one red})$.

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

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

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 $P(\text{defective})$.

Given $P(A) = 0.5$, $P(B) = 0.3$ and $P(A \cap B) = 0.15$, determine whether $A$ and $B$ are independent.

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

A box has 6 red and 4 blue balls. Three are drawn without replacement. Find $P(\text{all three red})$.

In a survey, $n(A) = 30$, $n(B) = 25$, $n(A \cap B) = 10$ and $n(U) = 60$. Find $P(A \mid B)$.

Given $P(A \cup B) = 0.7$, $P(B) = 0.4$ and $P(A \cap B) = 0.1$, find $P(A)$.

$3\%$ 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 $P(\text{positive test})$.

At a school, $50\%$ study French and $30\%$ study Spanish. Of those who study French, $20\%$ also study Spanish. Find $P(F \cap S)$.

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

$A$, $B$, $C$ are mutually independent with $P(A) = P(B) = P(C) = p$. Find $P(\text{exactly one of } A, B, C)$ when $p = \dfrac{1}{4}$.

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

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

A bag has 5 red, 2 blue and 3 green balls. Two are drawn without replacement. Find $P(\text{second is red} \mid \text{first is not red})$.

A shooter scores with probability $0.4$. How many shots so that $P(\text{at least one score}) \geq 0.95$?

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

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

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

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

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

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