Probability: Mathematics 9709 (Cambridge International AS & A Level)

In a school raffle, tickets are numbered from $1$ to $50$, and one winning ticket is drawn at random so that every number is equally likely.

What is the probability that the winning number is a multiple of $5$ or a multiple of $8$?

A fitness studio has $120$ members. Each member's attendance is recorded for two classes, Yoga and Spin, and summarised in the two-way table below.

One member is selected at random from the $120$ members.

(a) Find $P(\text{Yoga})$ and $P(\text{Yoga and Spin})$. [2]

(b) State, with a reason, whether the events "attends Yoga" and "attends Spin" are mutually exclusive. [1]

(c) Determine, showing your working, whether the events "attends Yoga" and "attends Spin" are independent. [3]

(d) Find $P(\text{Spin} \mid \text{Yoga})$. [2]

QuickParcel Couriers dispatches all of its parcels from one of two hubs. Hub A dispatches $60\%$ of all parcels, and Hub B dispatches the remaining $40\%$.

The probability that a parcel dispatched from Hub A arrives late is $0.05$, and the probability that a parcel dispatched from Hub B arrives late is $0.10$.

A parcel is selected at random.

(a) Find the probability that the parcel is dispatched from Hub A and arrives late. [2]

(b) Find the probability that the parcel arrives late. [3]

(c) Given that the parcel arrives late, find the probability that it was dispatched from Hub B. [3]

A warehouse has two smoke detectors fitted in the same storage room, Detector 1 and Detector 2, which operate independently of one another.

If a fire occurs, the probability that Detector 1 raises an alarm is $0.9$, and independently, the probability that Detector 2 raises an alarm is $0.8$.

Given that a fire occurs, find the probability that:

(a) both detectors raise the alarm; [2]

(b) neither detector raises the alarm; [2]

(c) at least one detector raises the alarm; [2]

(d) exactly one of the two detectors raises the alarm. [3]

For two events $A$ and $B$, it is given that $P(A) = 0.4$, $P(B) = 0.5$ and $P(A \cap B) = 0.2$.

Which one of the following statements is correct?

In a class of $40$ students, every student plays badminton, chess, both, or neither. The numbers in each category are shown in the table below.

A student is selected at random from the class.

Given that the student plays badminton, what is the probability that they also play chess?

A box contains $12$ chocolates: $7$ dark chocolates and $5$ milk chocolates, otherwise identical in appearance. Three chocolates are selected at random from the box, all at the same time, so the order of selection does not matter.

(a) Find the total number of ways to choose $3$ chocolates from the $12$, and hence find the probability that all three chocolates selected are dark. [3]

(b) Find the probability that exactly two of the three chocolates selected are dark (and one is milk). [3]

(c) Find the probability that at least one milk chocolate is selected. [3]

A bag contains $5$ red balls and $3$ blue balls, otherwise identical. Two balls are drawn at random from the bag, one after the other, without replacement.

(a) By considering the two stages of the draw, find $P(\text{both balls are red})$. [2]

(b) Find $P(\text{exactly one of the two balls is red})$. [3]

(c) Given that exactly one of the two balls drawn was red, find the probability that the red ball was drawn first. [3]

Two fair six-sided dice, each numbered $1$ to $6$, are rolled together. Let $A$ be the event "the sum of the two scores is $9$" and let $B$ be the event "the two dice show the same number (a double)".

What is $P(A \text{ or } B)$?

A magazine publisher surveys $200$ subscribers about whether they read Magazine $X$, Magazine $Y$, both, or neither. The results are shown in the table below.

A subscriber is selected at random.

(a) Find $P(X)$, $P(Y)$ and $P(X \cap Y)$. [2]

(b) Use the addition rule to find $P(X \cup Y)$, and hence find the probability that the subscriber reads neither magazine. [3]

(c) Determine, showing your working, whether the events "reads $X$" and "reads $Y$" are independent. [2]

(d) Given that the subscriber does not read Magazine $Y$, find the probability that they read Magazine $X$. [2]