Probability & Statistics 1 — Variant 2

George has a fair $5$-sided spinner with sides labelled $1, 2, 3, 4, 5$. He spins the spinner and notes the number on the side on which the spinner lands.

Find the probability that it takes fewer than $7$ spins for George to obtain a $5$. George spins the spinner $10$ times.

Find the probability that he obtains a $5$ more than $4$ times but fewer than $8$ times.

A factory produces a certain type of electrical component. It is known that $15\%$ of the components produced are faulty. A random sample of $200$ components is chosen. Use an approximation to find the probability that more than $40$ of these components are faulty.

The heights of the members of a club are normally distributed with mean $166$ cm and standard deviation $10$ cm.

Find the probability that a randomly chosen member of the club has height less than $170$ cm.

Given that $40\%$ of the members have heights greater than $h$ cm, find the value of $h$ correct to 2 decimal places.

The random variable $X$ is normally distributed with mean $\mu$ and standard deviation $\sigma$. Given that $\sigma = \frac{2}{3}\mu$, find the probability that a randomly chosen value of $X$ is positive.

Freddie has two bags of marbles. Bag $X$ contains $7$ red marbles and $3$ blue marbles. Bag $Y$ contains $4$ red marbles and $1$ blue marble. Freddie chooses one of the bags at random. A marble is removed at random from that bag and not replaced. A new red marble is now added to each bag. A second marble is then removed at random from the same bag that the first marble had been removed from.

Draw a tree diagram to represent this information, showing the probability on each of the branches.

Find the probability that both of the marbles removed from the bag are the same colour.

Find the probability that bag $Y$ is chosen given that the marbles removed are not both the same colour.

Find the number of different arrangements of the $9$ letters in the word ANDROMEDA in which no consonant is next to another consonant. (The letters D, M, N and R are consonants and the letters A, E and O are not consonants.)

Find the number of different arrangements of the $9$ letters in the word ANDROMEDA in which there is an A at each end and the Ds are not together. Four letters are selected at random from the $9$ letters in the word ANDROMEDA.

Find the probability that this selection contains at least one D and exactly one A.