Conditional probability and independent events: notes and practice questions

If there are two events A and B are its given that $P(A) = 0.5$ and $P(B) = 0.2$. let $P(AUB) = x.$

$a$ Find the value of $x$ if the events A and B are said to be mutually exclusive.

$b$ Find the value of $x$ if the events A and B are said to be independent.

If two events A and B are said to be independent events where $P(A) = 0.4$ and $P\left( A^{'} \cap B \right) = 0.3$. let $P(A \cap B) = x.$

$a$ Find the value of $x$.

$b$ Find $P(A|B').$

A spinner with four sectors is spun. The sectors are numbered 1, 2, 3, and 4. Let $X$ be the score obtained when the spinner is spun. The probability distribution for $X$ is given in the following table.

(a) Find the value of $k$.

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

A second spinner, B, is also spun. Let $Y$ be the score obtained. The probability distribution for $Y$ is given in the following table.

(c) (i) State the range of possible values of $n$.

(ii) Hence, find the range of possible values of $m$.

(d) Hence, find the range of possible values for $E(Y)$.

Leo spins spinner A once and Mia spins spinner B once. The probability that Leo's score is greater than Mia's score is $\frac{2}{5}$.

(e) Find the value of $E(Y)$.

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 biased four-sided spinner, A, is spun. Let $S_A$ be the score obtained. The probability distribution for $S_A$ is given in the following table.

Find the value of $k$.

Hence, find the value of $E(S_A)$.

A second biased four-sided spinner, B, is spun. Let $S_B$ be the score obtained. The probability distribution for $S_B$ is given in the following table.

(i) State the range of possible values of $b$.

(ii) Hence, find the range of possible values of $a$.

Hence, find the range of possible values for $E(S_B)$.

Leo spins spinner A once and Mia spins spinner B once. The probability that Leo's score is greater than Mia's score is $\frac{13}{49}$.

Find the value of $E(S_B)$.

A factory produces two types of light bulbs: standard and long-life. The lifespan of the bulbs, in hours, can be modelled as normal distributions with the following parameters.

(a) Find the percentage of standard bulbs that have a lifespan of less than 700 hours.

(b) The factory produces a large number of bulbs, of which 60% are standard bulbs. Both types of bulbs are produced and randomly mixed together for packaging. A quality control process identifies and removes all bulbs with a lifespan of less than 700 hours. An inspector randomly selects a bulb from this removed group. Find the probability that it is a standard bulb.

A student, Chloe, travels to school by bus. She can take one of two routes, Route X or Route Y. She is equally likely to choose either route on any given day.

The probability that she arrives on time when taking Route X is $\frac{3}{4}$.

The probability that she arrives on time when taking Route Y is $\frac{2}{5}$.

(a) Find the probability that Chloe arrives on time for school on a randomly chosen day.

Let $X$ be the event that Chloe chooses Route X and let $T$ be the event that she arrives on time.

(b) Determine whether events $X$ and $T$ are independent.

(a) The random variable $X$ follows a normal distribution with mean $\mu$ and standard deviation $\sigma$.

Find $P(\mu - 1.2\sigma < X < \mu + 1.2\sigma)$.

(b) The diameters of ball bearings produced by a factory, in mm, are normally distributed with mean $\mu$ and standard deviation $\sigma$. The ball bearings are categorized as defective, standard, large, or premium, according to their diameter. The following table shows the probability a ball bearing is classified into each category.

The maximum diameter of a defective ball bearing is 14.8 mm.

The minimum diameter of a premium ball bearing is 16.5 mm.

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

(c) The factory rejects all defective ball bearings. The remaining ball bearings are sold.

Find the probability that a ball bearing chosen at random from those sold is categorized as

(i) standard;

(ii) large;

(iii) premium.

(d) The selling prices of the different categories of ball bearings at this factory are shown in the following table:

The factory incurs a fixed cost of $300 for the production run and assumes it will sell the accepted ball bearings in exactly the same proportion as calculated in part (c).

According to this model, find the minimum number of accepted ball bearings that must be sold so that the net profit for the factory is at least $550.

(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 $\sigma$.

Two archers, Clara and David, are competing. To decide who shoots an arrow, a fair six-sided die is rolled. If the die shows a 1, Clara is chosen. If the die shows any other number, David is chosen.

The probability that Clara hits the target is $\frac{4}{5}$. The probability that David hits the target is $\frac{2}{3}$.

(a) Find the probability that the target is hit.

(b) Let $C$ be the event that Clara is chosen and let $H$ be the event that the target is hit. Determine, with a reason, whether events $C$ and $H$ are independent.

(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.

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.

Consider events $C$ and $D$ such that $P(C') = \frac{2}{3}$, $P(C \cup D) = \frac{2}{3}$ and $P(D|C) = \frac{1}{2}$.

(a) Find $P(C \cap D)$.

(b) Determine if events $C$ and $D$ are independent. Justify your answer.

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 lifespan of a certain type of rechargeable battery, in hours, can be modelled by a normal distribution with a mean of 1500 hours and a standard deviation of 50 hours. A battery is deemed faulty and rejected if its lifespan is less than 1425 hours.

Find the probability that a randomly selected battery is rejected.

(b) Estimate the number of batteries that will be rejected from a random sample of 200 batteries.

(c) Given that a battery is not rejected, find the probability that it has a lifespan greater than 1575 hours.

(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 $\sigma$.

At a tech company, employees can participate in various initiatives. 70% of employees attend a professional development workshop ($W$), and 20% of employees work on a special innovation project ($P$). 20% of employees do neither activity.

An employee is selected at random.

Find the probability that the employee attends a professional development workshop and works on a special innovation project.

Find the probability that the employee works on a special innovation project, but does not attend a professional development workshop.

At the company, 40% of the employees are in the Marketing department. Of those in the Marketing department, 30% work on a special innovation project.

An employee is selected at random. Let $M$ be the event "the employee is in the Marketing department" and let $P$ be the event "the employee works on a special innovation project".

Find $P(M \cap P)$.

Determine if the events $M$ and $P$ are independent. Justify your answer.