Binomial distribution: notes and practice questions

A manufacturer produces electronic components. It is known that the probability that a randomly selected component is defective is 0.03. A quality control inspector takes a random sample of 40 components from a large batch.

(a) Find the probability that there is at least one defective component in the sample.

(b) Given that there is at least one defective component in the sample, find the probability that there are at most three defective components.

(a) An electronics factory produces two types of resistors: Type A and Type B.

The resistance, $R_A$ (in Ohms), of Type A resistors is normally distributed with a mean of 120 Ohms and a standard deviation of 5 Ohms.

Find the probability that a randomly selected Type A resistor has a resistance less than 115 Ohms.

(b) In a random selection of 10 Type A resistors, find the probability that exactly 3 have a resistance less than 115 Ohms.

(c) The resistance, $R_B$ (in Ohms), of Type B resistors is normally distributed with a mean of 135 Ohms and a standard deviation of 7 Ohms.

Each day, 70% of the resistors produced are Type A, and 30% are Type B.

On a particular day, a resistor is randomly selected from all those produced at the factory.

Let $A$ represent 'Type A resistor' and $B$ represent 'Type B resistor'.

(i) Find the probability that the randomly selected resistor has a resistance less than 115 Ohms.

(ii) Given that a randomly selected resistor has a resistance less than 115 Ohms, find the probability that it is a Type A resistor.

(d) The machine that makes the Type A resistors is adjusted so that the mean resistance of the Type A resistors remains the same, but their standard deviation changes to $\sigma$ Ohms. The machine that makes the Type B resistors is not adjusted. The probability that the resistance of a randomly selected resistor from these machines is now less than 115 Ohms is 0.18.

Find the value of $\sigma$.

The delivery times, $T$ minutes, for packages from a logistics hub to a regional distribution center can be modelled by a normal distribution with a mean of 120 minutes and a standard deviation of $\sigma$ minutes.

Given that 3% of the delivery times are longer than 135 minutes, find the value of $\sigma$.

Find the probability that a randomly selected package will have a delivery time of more than 130 minutes.

Given that a package delivery takes longer than 130 minutes, find the probability that it takes less than 135 minutes.

On a particular day, there are 80 packages scheduled for delivery from the hub.

Find the expected number of packages that will have a delivery time of more than 130 minutes.

Find the probability that more than 8 of the packages on this particular day will have a delivery time of more than 130 minutes.

(a) A new automated coffee machine is programmed to dispense coffee. The volume of coffee dispensed, $V$ ml, is normally distributed with a mean of 200 ml and a standard deviation of $\sigma$ ml.

On 15% of occasions, the machine dispenses more than 210 ml of coffee.

Find the value of $\sigma$.

(b) On a randomly selected occasion, find the probability that the machine dispenses more than 205 ml of coffee.

(c) The machine is considered to have 'over-filled' a cup if it dispenses more than 215 ml of coffee. Seven customers order coffee. Assume the volume dispensed for each customer is independent.

Find the probability that at least one of these seven coffees is over-filled.

(d) Given that at least one of the seven coffees is over-filled, find the probability that exactly two of them are over-filled.

(e) The café serves 25 customers in an hour. The machine requires maintenance if it over-fills more than 3 coffees during that hour. So far, 18 customers have been served, and the machine has over-filled 2 coffees.

Find the probability that the machine will NOT require maintenance by the end of the hour.

(a) The lifespan of a new type of battery, $L$ hours, can be modelled by a normal distribution with a mean of 1200 hours and a standard deviation of $\sigma$ hours.

Given that 5% of the batteries last longer than 1280 hours, find the value of $\sigma$.

(b) Find the probability that a randomly selected battery will have a lifespan of less than 1150 hours.

(c) Given that a battery lasts longer than 1150 hours, find the probability that it lasts less than 1250 hours.

(d) A batch of 500 batteries is produced. Find the expected number of batteries that will have a lifespan of less than 1150 hours.

(e) Find the probability that more than 85 of the batteries in this batch will have a lifespan of less than 1150 hours.

In a large university, 200 students were surveyed. Of those, 120 were undergraduates (U) and the rest postgraduates (P).

Each student in the survey was asked whether they preferred quiet zones (Q) or collaborative areas (C) for studying. It was found that 75 of the undergraduates preferred quiet zones. The total number of students who preferred collaborative areas was 100. This information is shown in the following table.

Find the value of

$p$;

$q$.

Three students are chosen at random from those surveyed. Find the probability that all three are postgraduates.

Given that $P(P|Q) = \frac{1}{4}$, find the value of $x$.

A student is chosen at random from those surveyed. Write down the probability that they are a postgraduate who prefers quiet zones.

Determine if the events P (Postgraduate) and Q (prefers Quiet Zones) are independent. Justify your answer.

It can be assumed that the survey results are representative of the university population. Ten students from the university are chosen at random. Find the probability that at least five of them prefer quiet zones.

The lifespan, $L$ (in hours), of a certain type of LED light bulb is modelled by a normal distribution with mean $\mu$ and standard deviation $\sigma$.

It is known that $P(L < 18500) = 0.15$ and $P(L > 21500) = 0.30$.

Find the probability that a randomly selected light bulb has a lifespan between $18500$ hours and $21500$ hours.

Find the value of $\mu$ and the value of $\sigma$.

A manufacturer tests a batch of $80$ randomly selected light bulbs. Any bulb with a lifespan greater than $21500$ hours is considered a 'long-life' bulb. Lifespans of bulbs are independent of each other.

Find the probability that exactly $25$ bulbs in the batch are 'long-life' bulbs.

Given that fewer than $30$ bulbs are 'long-life' bulbs, find the probability that exactly $25$ bulbs are 'long-life' bulbs.

In another factory, a different type of LED light bulb is produced. The lifespan of these bulbs, $F$ (in hours), is normally distributed with a mean of $22000$ hours. The interquartile range (IQR) for these bulbs is $3000$ hours.

Find the value of the standard deviation, $d$, for this type of bulb.

(a) The volume, $V$ ml, of liquid in bottles filled by a machine can be modelled by a normal distribution with mean $500$ ml and standard deviation $8$ ml.

A bottle is selected at random.

Find the probability that it contains more than $510$ ml.

(b) According to this model, $75\%$ of the bottles contain between $w$ ml and $510$ ml.

Find the probability that a randomly selected bottle contains less than $w$ ml.

(c) Find the value of $w$.

(d) A quality control inspector randomly selects $12$ bottles.

Find the probability that exactly $2$ of these bottles contain less than $w$ ml.

(a) The resistance, R ohms, of resistors produced by a factory is normally distributed with a mean of 100 ohms and a standard deviation of 3.5 ohms.

Find the probability that a randomly selected resistor has a resistance less than 98 ohms.

(b) In a random sample of 15 resistors, find the probability that exactly 4 of them have a resistance less than 98 ohms.

(c.i) The capacitance, C microfarads, of capacitors produced by the same factory is normally distributed with a mean of 50 $\mu$F and a standard deviation of 2.8 $\mu$F. Each day, 70% of the components produced are resistors and 30% are capacitors.

Find the probability that a randomly selected component has a value less than its respective threshold (i.e., less than 98 ohms for a resistor or less than 47 $\mu$F for a capacitor).

(c.ii) Given that a randomly selected component has a value less than its respective threshold, find the probability that it is a resistor.

(d) The resistor manufacturing process is adjusted so that the mean resistance remains 100 ohms but its standard deviation changes to $\sigma$ ohms. The capacitor manufacturing process is not adjusted. The probability that a randomly selected component from these machines has a value less than its respective threshold is now 0.160.

Find the value of $\sigma$.

A quality control manager inspects a batch of 40 components for defects. The number of defective components, $X$, follows a binomial distribution, $X \sim B(40, p)$. The variance of the number of defective components is known to be 8.4.

(a) Find the possible values of $p$.

The cost of repairing the batch of components, $Y$, is given by the formula $Y = 7 - 4X$ (in dollars).

(b) Find Var($Y$).

A new automated manufacturing process produces components. The time, in minutes, taken for a critical assembly step is modelled by a continuous random variable $X$, with a probability density function defined by

$f(x) = \begin{cases} \frac{2}{\pi \sqrt{9-x^2}} & 0 \leq x \leq 3 \\ 0, & \text{otherwise.} \end{cases}$

Find the exact value of $E(X)$.

Find $P(X < 1.5)$.

The assembly step is considered "efficient" if it takes less than 1.5 minutes. Each assembly step is independent. Determine the least number of assembly steps required to be 99% sure of at least one efficient step.

Ten assembly steps were conducted.

Find the probability that exactly three steps were efficient.

Write down the number of ways these three efficient steps could have occurred consecutively in a batch of 10.

Now consider a batch of $n$ assembly steps where it is given that exactly three efficient steps have occurred.

Write down an expression for the number of ways these three efficient steps could have occurred consecutively.

Find the greatest value of $n$ such that the probability of three consecutive efficient steps is more than 0.05, given that exactly three efficient steps have occurred in the batch.

A textile factory produces rolls of fabric. Due to manufacturing imperfections, $7\%$ of the fabric rolls produced are found to have minor defects. A quality control inspector randomly selects a batch of $30$ fabric rolls for inspection.

(a) Find the probability that exactly two of the selected fabric rolls have minor defects.

(b) Find the probability that no more than three of the selected fabric rolls have minor defects.

(c) Find the probability that at least two of the selected fabric rolls have minor defects.

(d) Find the variance of the number of fabric rolls with minor defects in a batch of $30$.

A discrete random variable $X$ follows a binomial distribution, $X \sim B(n, \frac{1}{3})$.

It is given that the probabilities $P(X=r-1)$, $P(X=r)$ and $P(X=r+1)$ for some integer $r$ form an arithmetic sequence, where $1 \le r \le n-1$.

(a) Show that $n^2 - 3n(2r+1) + 9r^2 + 3r - 4 = 0$.

(b) For a particular experiment, it is known that $n=8$. Find the possible value(s) of $r$.

A telemarketing company is launching a new campaign. Based on previous data, the probability that a single call results in a successful sale is $0.15$. A sales representative makes $25$ calls each week, and the outcome of each call is independent.

(a) Find the probability that the sales representative makes exactly $5$ successful sales in a given week.

(b) Find the probability that the sales representative makes at least $4$ successful sales in a given week.

(c) Find the probability that the first successful sale occurs on the $8$th call.

The mass, $M$ grams, of mangoes grown in an orchard can be modelled by a normal distribution with mean $\mu$ and standard deviation $\sigma$. The masses of all mangoes are independent of each other.

$4.2\%$ of the mangoes have a mass of less than $240$ grams, while $9.5\%$ of the mangoes have a mass of more than $380$ grams.

Find the value of $\mu$ and the value of $\sigma$.

Hence, find the probability that a mango chosen at random has a mass of between $270$ and $340$ grams.

The farmer also grows avocados. Records show that $74.8\%$ of the avocados grown are classified as large.

The probability of an avocado being classified as large is independent of any other avocado.

On a particular day, $24$ avocados are randomly selected.

Find the probability that at least $15$ of these avocados are classified as large.

Given that at least $15$ of these avocados are classified as large, find the probability that more than $4$ are not classified as large.

Leo has a box of tokens which are coloured either silver, gold or bronze.

The box contains exactly 15 bronze tokens. The number of silver tokens is four times the number of gold tokens.

Leo plays a game where he takes 12 tokens out of the box, one at a time. He notes the colour of each token and returns it to the box before taking the next token out of the box.

The probability that the first token is bronze is 0.3.

Show that there are 28 silver tokens in the box.

Find the probability that at least 7 of the 12 tokens that Leo takes are bronze. Give your answer correct to five significant figures.

Leo has to pay $\$10$ to take part in the game. If he takes at least 7 tokens of the same colour, he wins a prize. If he does not take at least 7 tokens of the same colour, then he does not win a prize. There is a different prize for each colour, as shown in the following table, where $C \in \mathbb{Z}^+$.

Let the random variable $X$ represent Leo's net gain in dollars when he plays the game once. For example, if he takes at least 7 gold tokens, his net gain is $100 - 10 = 90$ since he gains $\$90$.

The probability distribution of $X$ is shown in the following table, with probabilities given correct to four decimal places, where $A \in \mathbb{Z}, q \in \mathbb{R}$.

Write down the value of $A$.

Use the probabilities in the table to find the value of $q$.

Determine the smallest integer value of $C$ for which Leo could expect to make a positive net gain.

Leo wants to play the game until he wins a prize.

Find the minimum number of times Leo needs to play the game in order that the probability of winning at least one prize is greater than 0.999.