Statistics and probability: 197 IB Maths AA HL practice questions

At a skiing competition the mean time of the first three skiers is 34.1 seconds. The time for the fourth skier is then recorded and the mean time of the first four skiers is 35.0 seconds. Find the time achieved by the fourth skier.[4]

There are 75 players in a golf club who take part in a golf tournament. The scores obtained on the 18th hole are as shown in the following table.

(a) One of the players is chosen at random. Find the probability that this player's score was 5 or more.[2]

(b) Calculate the mean score.[2]

A mathematics test is given to a class of 20 students. One student scores 0, but all the other students score 10.

(a) Find the mean score for the class.[2]

(b) Write down the median score.[1]

(c) Write down the number of students who scored

(i) above the mean score;[2]

(ii) below the median score.

Events $A$ and $B$ are such that P$(A) = 0.65$, P$(B) = 0.75$ and P$(A \cap B) = 0.6$.

(a) Find P$(A \cup B)$.[2]

(b) Hence, or otherwise, find P$(A' \cap B')$.[2]

The marks obtained by a group of students in a class test are shown below.

Given the mean of the marks is 6.5, find the value of $k$.[4]

Let $A$ and $B$ be events such that $P(A) = 0.5$, $P(B) = 0.4$ and $P(A \cup B) = 0.6$. Find $P(A \mid B)$.[5]

It is believed that the lifespans of Manx cats are normally distributed with a mean of 13.5 years and a variance of 9.5 years$^2$.

(a) Calculate the range of lifespans of Manx cats whose lifespans are within one standard deviation of the mean.[2]

(b) Estimate the number of Manx cats in a population of 10 000 that will have a lifespan of less than 10 years. Give your answer to the nearest whole number.[3]

A set of 15 observations has mean 11.5 and variance 9.3. One observation of 22.1 is considered unreliable and is removed. Find the mean and variance of the remaining 14 observations.[6]

The number of hours spent exercising each week by a group of students is shown in the following table.

The median is 4.5 hours.

(a) Find the value of $x$.[2]

(b) Find the standard deviation.[2]

The number of bananas that Lucca eats during any particular day follows a Poisson distribution with mean $0.2$.

(a) Find the probability that Lucca eats at least one banana in a particular day.[2]

(b) Find the expected number of weeks in the year in which Lucca eats no bananas.[4]

A discrete random variable $X$ has the probability distribution given by the following table.

Given that $\text{E}(X) = \frac{19}{12}$, determine the value of $p$ and the value of $q$.[4]

Events $A$ and $B$ are such that P$(A) = 0.2$ and P$(B) = 0.5$.

(a) Determine the value of P$(A \cup B)$ when

(i) $A$ and $B$ are mutually exclusive;[2]

(ii) $A$ and $B$ are independent.[2]

(b) Determine the range of possible values of P$(A | B)$.[3]

The random variables $X$ and $Y$ are normally distributed with $X \sim N(7, a^2)$ and $Y \sim N(19, a^2)$, where $a > 0$.

(a) Find $b$ such that $\mathrm{P}(X > b) = \mathrm{P}(Y > 22)$.[2]

(b) Write down the approximate value of $\mathrm{P}(7 - a < X < 7 + a)$, correct to two significant figures.[1]

(c) Given that $a = 3$, calculate the approximate value of $\mathrm{P}(Y < 22)$, correct to two significant figures.[3]

A team of 6 players is to be selected from 10 volleyball players, of whom 8 are boys and 2 are girls.

(a) In how many ways can the team be selected?[2]

(b) In how many of these selections is exactly one girl in the team?[3]

(c) If the selection of the team is made at random, find the probability that exactly one girl is in the team.[2]

The random variable $X$ has the distribution $\text{B}(30, p)$. Given that $\text{E}(X) = 10$, find

(a) the value of $p$;[1]

(b) P$(X = 10)$;[2]

(c) P$(X \geq 15)$.[2]