Venn diagrams, tree diagrams, probability tables (+ notation & combined & mutually exclusive events): notes and practice questions

A bakery produced a batch of 200 loaves of bread, consisting of sourdough and rye. The sales outcomes for these loaves are shown in the following table.

(a) Find the probability that a randomly chosen loaf from this batch was sold by the bakery.

A loaf is chosen at random from this batch. It is found that this loaf was sold.

(b) Find the probability that the loaf was a sourdough loaf.

Two different loaves are chosen at random from the original batch of 200 loaves.

(c) Find the probability that both loaves were rye.

A board game involves drawing cards from a special deck. Each card has a score printed on it. The possible scores are $-5, -2, 0, 3, 4, 6$. The following table shows the probability distribution for the score, $X$, when a card is drawn.

Score $x$

$-5$

$-2$

$0$

$3$

$4$

$6$

$P(X=x)$

$\frac{1}{15}$

$p$

$\frac{2}{15}$

$\frac{3}{15}$

$\frac{4}{15}$

$\frac{1}{15}$

(a) Find the exact value of $p$.

(b) Calculate the expected score when drawing one card.

A player draws two cards from the deck, replacing the first card before drawing the second. Find the probability that the total score from the two cards is $1$.

A game involves two stages. First, a player draws a marble from a bag containing 3 red marbles and 2 blue marbles. After drawing a marble, the player spins a fair three-sided spinner with sections labeled 1, 2, and 3.

If a red marble is drawn, the player's final score is the number shown on the spinner.

If a blue marble is drawn, the player's final score is two more than the number shown on the spinner.

Find the probability that a player's final score is 4.

Complete the following table, showing the probability distribution of the final score.

Calculate the expected value of the player's final score.

A manufacturing company inspects the first 50 items produced each morning for defects.

(a) State the sampling method being used.

The company uses an automated machine to test products for defects. This machine is not perfect.

It is known that 3% of all products manufactured are defective (D).

If a product is defective, the machine correctly identifies it as defective (tests positive, $T+$) 98% of the time.

If a product is not defective (D'), the machine incorrectly identifies it as defective (tests positive, $T+$) 1% of the time.

The tree diagram shows some of this information.

(b) (i) Write down the value of $P(D')$.

(ii) Write down the value of $P(T-|D)$.

(iii) Write down the value of $P(T+|D')$.

(iv) Write down the value of $P(T-|D')$.

(c) Use the tree diagram to find the probability that a randomly selected product:

(i) is not defective and tests positive.

(ii) tests negative.

(iii) is defective given that it tested negative.

(d) The company finds the actual number of defective products in their sample is different than predicted by the tree diagram. Explain why this might be the case.

(e) The factory manager surveyed all employees on a particular shift. All employees on this shift worked in at least one of these departments: Assembly (A), Painting (P), or Quality Control (Q). It was found that:

• 85 employees worked in Assembly;
• 55 employees worked in Painting;
• 35 employees worked in Quality Control;
• 10 employees worked in all three departments;
• 20 employees worked in Assembly and Quality Control but not Painting;
• 15 employees worked in Assembly and Painting but not Quality Control;
• 4 employees worked only in Quality Control.

Draw a Venn diagram to illustrate this information, placing all relevant information on the diagram.

(f) Find the total number of employees on this shift.

The diameter of a certain type of industrial component is modelled by a normal distribution with a mean of $50$ mm and a standard deviation of $0.5$ mm.

Find the probability that a randomly selected component has a diameter less than $50.7$ mm.

Find the probability that a randomly selected component has a diameter greater than $51.2$ mm.

Assuming the diameters of components are independent, find the probability that two consecutive components both have a diameter greater than $51.2$ mm.

A component is classified as 'premium' if its diameter is between $49.5$ mm and $50.5$ mm. A batch of three components is considered 'high quality' if all three components in the batch are premium. Find the probability that a randomly selected batch is NOT high quality.

In a production run, $12$ batches of three components are produced. Find the probability that at least $7$ of these batches are high quality.

Find the probability that between $7$ and $10$ (exclusive of $10$) of these batches are high quality.

Given that at least $7$ batches are high quality, find the probability that less than $10$ batches are high quality.

In a survey of students at a local college, it was found that the probability a student studies Art (event $A$) is $P(A) = 0.4$. The probability a student studies Biology (event $B$) is $P(B) = 0.7$. The probability that a student studies either Art or Biology or both is $P(A \cup B) = 0.8$.

(a) Calculate the probability that a randomly selected student studies Biology but not Art.

(b) Determine the probability that a randomly selected student studies Art or does not study Biology.

(c) Find the probability that a randomly selected student does not study both Art and Biology.

A company manufactures custom-designed shelving units. The number of shelf segments in a unit is determined by a "tier number", $N_k$. The $k$th tier number, $N_k$, represents the total number of individual shelf segments required to build a triangular display unit with $k$ levels, and is defined as $N_k = \sum_{r=1}^k r$.

Calculate the total number of shelf segments required for a unit with 6 levels, $N_6$.

Determine the formula for $N_k$ in the form $ak^2 + bk$.

A designer combines two units: one with 5 levels and another with 4 levels. Find the total number of shelf segments needed, $N_5 + N_4$.

Find the simplest expression for the total number of shelf segments required if a designer combines a unit with $k$ levels and another with $k-1$ levels, i.e., $N_k + N_{k-1}$.

In a quality control batch, there are 15 standard shelf segments and 10 reinforced shelf segments. Two segments are chosen at random from the batch without replacement.

Calculate the probability that the two segments are of different types (one standard, one reinforced).

A new batch of shelf segments contains $N_k$ standard segments and $N_{k-1}$ reinforced segments. Two segments are chosen at random from the batch without replacement.

Show that the probability that the two segments are of different types is independent of $k$.

In a survey of $100$ students at a high school, data was collected on their subject choices. It was found that $65$ students were studying Mathematics, and $45$ students were studying Physics.

Every student surveyed was studying at least one of these two subjects.

(a) Determine how many students were studying both Mathematics and Physics.

(b) Find the probability that a randomly selected student studies Mathematics, but not Physics.

(c) Explain why the events "studying Mathematics" and "studying Physics" are not independent events.

Tech Solutions Inc. is a customer support company. They are analyzing their call center efficiency. Over a long period, they collect data on the number of calls, $C$, already in the queue when a new customer's call arrives. The probability distribution of $C$ is shown in the following table.

Find the probability that there are at least two calls in the queue when a new customer's call arrives.

Find E($C$).

The time in seconds, $T$, taken to resolve a single customer's issue can be modelled by the normal distribution $T \sim \text{N}(130, 25^2)$.

The company's management estimates that the expected total time a new customer will wait before their issue is resolved can be found by calculating E($C$) $\times$ E($T$).

Find the value of E($C$) $\times$ E($T$).

The company considers a service time to be 'long' if it takes more than three minutes to resolve a single customer's issue.

Using the distribution of $T$ given above, find the probability that it takes more than three minutes to resolve a randomly selected customer's issue.

Find the probability it takes more than four minutes in total to resolve two randomly selected customers' issues. You may assume all service times are independent of all other service times.

The company assumes that when a new customer's call arrives, the person at the front of the queue has only just reached a support agent. They also assume that if there are three or more customers already in the queue, the new customer will definitely wait more than three minutes before being served.

Using these assumptions and the probabilities for $C$ given in the table above,

find the probability a customer just arriving at the call center will wait more than three minutes before being served.

The company has a policy to employ more staff if the probability that a customer has to wait more than three minutes before being served is greater than $0.35$.

Hence state whether Tech Solutions Inc. will decide to employ more staff.

A group of students were surveyed about their participation in three extracurricular clubs: Environmental Club (E), Music Club (M), and Theater Club (T). The probabilities of their participation are represented in the Venn diagram below.

Justify that events M and T are not independent.

Explain why events E and T are mutually exclusive.

Determine whether events E and M are independent.

Determine whether events E' and M' are mutually exclusive.

Find P(T $\cap$ E').

A quality control team at a manufacturing plant inspects newly produced widgets. The probability that a single widget has a detectable defect is $0.15$. The inspections are independent events.

(a) Find an expression for the probability that at least one defect is detected in $n$ inspections.

(b) Hence, determine the least number of inspections required for the probability of detecting at least one defect to be greater than $99.9\%$.

A factory produces two types of electronic components: Type A and Type B. $70\%$ of the components produced are Type A, and the remaining $30\%$ are Type B. The probability that a Type A component is defective is $0.08$, and the probability that a Type B component is defective is $0.12$. A component is chosen at random from the production line.

(a) Calculate the probability that the chosen component is defective.

(b) Calculate the probability that the chosen component is Type A and is defective.

(c) Calculate the probability that the chosen component is Type B or is defective.

A new mobile game, "Fantasy Quest", involves players drawing two cards to determine their turn's action and reward.

• Action Card Deck: Contains three types of cards: 'Attack' (A), 'Defend' (D), and 'Heal' (H). The probabilities are: P(A) = 0.5, P(D) = 0.3, P(H) = 0.2.

• Reward Card Deck: Contains two types of cards: 'Gold' (G) and 'Experience Points' (XP). The probabilities are: P(G) = 0.6, P(XP) = 0.4.

A player draws one card from each deck. The outcome of a turn is the combination of the Action card and the Reward card. Assume the draws are independent.

(a) Construct a probability table showing all possible outcomes and their associated probabilities.

(b) Calculate the probability that a player draws an 'Attack' card and receives 'Gold'.

(c) Calculate the probability that a player's turn involves either a 'Defend' action or receiving 'XP' (or both).

A collector has a box containing 20 vintage stamps. Of these, 4 are considered rare editions.

Three stamps are randomly selected from the box.

(a) Find the probability that exactly one of the selected stamps is a rare edition.

(b) Find the probability that at least one of the selected stamps is a rare edition.

A museum curator is selecting items for a new exhibit. From a collection of $20$ artifacts, $5$ are classified as ancient, and the remaining $15$ are modern. The curator randomly selects $2$ artifacts from the collection.

(a) Find the probability that exactly one of the selected artifacts is ancient.

(b) Find the probability that at least one of the selected artifacts is ancient.

In a survey, $60$ students were asked about their participation in extracurricular clubs. It was found that $40$ students were members of the Robotics Club, and $35$ students were members of the Chess Club.

Every student surveyed was a member of at least one of these two clubs.

(a) Determine how many students were members of both the Robotics Club and the Chess Club.

(b) Find the probability that a randomly selected student was a member of the Robotics Club, but not the Chess Club.

(c) Explain why the events "being a member of the Robotics Club" and "being a member of the Chess Club" are not independent events.

A factory produces two types of electronic components: Type A and Type B. $70\%$ of the components produced are Type A, and $30\%$ are Type B. The probability that a Type A component passes a quality control test is $0.92$, and the probability that a Type B component passes is $0.85$. A component is chosen at random from the production line.

(a) Calculate the probability that the component passes the quality control test.

(b) Calculate the probability that the component is Type A and passes the quality control test.

(c) Calculate the probability that the component is Type B or it passes the quality control test.

The discrete random variable $X$ represents the number of successful deliveries a courier makes in an hour. The probability distribution of $X$ is given by the following table:

$\begin{array}{|c|c|c|c|c|c|} \hline x & 0 & 1 & 2 & 3 & 4 \\ \hline P(X=x) & 0.1 & p+q & 0.2 & p-q & p+2q \\ \hline \end{array}$

Given that $E(X) = 2.2$, find the values of $p$ and $q$.

Find $P(X = 3|X \ge 1)$.