Probability basics (expected #, complementary events, probability of event): notes and practice questions

Let C be the event that a randomly chosen household in a town has a cat, and D be the event that the household has a dog. It is known that $P(C) = 0.4$, $P(D) = 0.5$ and $P(C \cap D) = 0.15$.

(a) Find the probability that a randomly chosen household has a cat or a dog.

(b) Find the probability that a randomly chosen household has neither a cat nor a dog.

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 $E(Y)$.

In a survey of a group of high school students, it was found that the probability that a student participates in the school's music program is 0.5, and the probability that a student plays on a school sports team is 0.6. The probability that a student participates in both is 0.2.

Let M be the event that a student participates in the music program and S be the event that a student plays on a sports team.

(a) Find the probability that a randomly selected student participates in the music program or plays on a sports team.

(b) Hence, find the probability that a randomly selected student participates in neither the music program nor a sports team.

On a Saturday at a cinema, a sample of 50 customers was randomly selected. They were asked how many snack items they had purchased. This information is summarized in the following frequency table.

It can be assumed that this sample is representative of all customers for the following day.

For the following day, Sunday, estimate

(i) the probability that a randomly selected customer will purchase at least one snack item.

For the following day, Sunday, estimate

(ii) the expected number of snacks purchased by a customer.

It is known that 800 customers will attend the cinema on Sunday. The average price of a snack item is $5.50.

Estimate the total revenue from snack sales on Sunday.

(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.

In a survey, a group of students were asked if they play football (F) or basketball (B). The probability that a student plays football is 0.6, and the probability that a student plays basketball is 0.7. The probability that a student plays at least one of these sports is 0.9.

(a) Find the probability that a student plays both football and basketball.

(b) Find the probability that a student plays neither football nor basketball.

A tech company is testing the battery life of its new smartphone. A sample of 120 phones are tested to see how long their batteries last under continuous video playback. The results are shown in the cumulative frequency graph below.

(a) Find the median battery life.

(b) The lowest 25% of battery lives in the sample are less than $k$ hours. Find the value of $k$.

(c) The same data is represented by the following frequency table.

Find the value of $p$ and the value of $q$.

(d) The company manufactures a batch of 10,000 of these smartphones. Estimate the number of phones in the batch that will have a battery life of more than 10 hours.

(e) The company wishes to advertise the 'typical' battery life of the phone based on this test.

(i) Explain why this testing method might not provide an accurate representation of the battery life for a typical user.

(ii) Suggest a more appropriate method for testing the battery life to represent a typical user.

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.

A card is drawn at random from a standard deck of 52 playing cards.

(a) Find the probability that the card is a Queen.

(b) Find the probability that the card is a spade.

(c) Find the probability that the card is a red card.

(d) Find the probability that the card is an even numbered card (2, 4, 6, 8, 10).

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 batch of $N$ electronic components contains $D$ defective components and $F$ functional components. Components are selected randomly, one by one, without replacement.

(i) Find, in terms of $N$, the probability that the first component selected is defective.

(ii) Given that $N = 25$, $D = 10$, and $F = 15$, find the probability that the first two components selected are defective.

(b) Show that the probability that the first two components selected are functional is 0.35.

(c) Find the probability that the first three components selected are all functional.

(d) Find the probability that at least one of the first three components selected is defective.

(e) A technician earns 10 points if the first defective component is found on the third test, and 50 points if the first defective component is found on the fourth test. The technician tests $k$ such batches. Find the least value of $k$ such that the technician's expected total score is greater than 100.

A fair spinner with 8 equal sectors, numbered 1 to 8, is spun once.

(a) Find the probability that the spinner lands on the number 7.

(b) Find the probability that the spinner lands on a multiple of 3.

(c) Find the probability that the spinner lands on a perfect square.

(d) Find the probability that the spinner lands on a number that is a factor of 8.

(e) Find the probability that the spinner lands on a number greater than 8.

A local coffee shop, "The Daily Grind", surveyed a random sample of 50 customers about their purchasing habits over one week. The following table shows the number of coffees purchased by these customers.

This sample is considered representative of all customers for the following week.

For the following week, estimate:

(a.i) the probability that a randomly selected customer will purchase at least one coffee;

(a.ii) the expected number of coffees a customer will purchase.

(b) The coffee shop expects to serve 800 customers in the following week. Each large batch of coffee brewed can serve a maximum of 20 cups.

Estimate the minimum number of large batches of coffee that must be brewed to meet the expected demand.

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 machine in a factory produces light bulbs. On average, 1 out of every 8 light bulbs is defective.

(a) The machine produces 400 light bulbs in a day. Calculate the expected number of defective light bulbs produced in one day.

(b) In a week, the machine produces 2800 light bulbs. Calculate the expected number of non-defective light bulbs produced in that week.

(c) The factory needs to ship an order of 1400 non-defective light bulbs. Calculate the total number of light bulbs the machine would be expected to produce to meet this order.

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.

(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) A beverage company fills bottles with juice. The volume of juice, in millilitres (ml), can be modelled by a normal distribution with a mean of 750 ml and a standard deviation of 2.5 ml. A bottle is considered underfilled and rejected if its volume is less than 746 ml.

Find the probability that a randomly selected bottle is rejected.

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

(c) Given that a bottle is not rejected, find the probability that its volume is greater than 753 ml.