BUSINESS MATHEMATICS NEB 12 | OLD IS GOLD EXPLANATION OF QUESTIONS AND SOLUTIONS STEPS BY STEPS

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Series: Business Mathematics - Old is Gold (Derivatives)

Question No: 1

Topic: Application of Derivatives (Elasticity of Demand)

Marks: 5

Question: The demand for a commodity is given by Q = 48 - 3P².
(a) Find the point elasticity of demand in terms of P.
(b) Find the elasticity when P = 3.
(c) If the price of 3 is decreased by 4%, find the percentage increase in demand.

1. Explanation of the Question

This is a multi-part case question that tests your understanding of how derivatives apply to economics. First, you are asked to derive the general formula for point elasticity (E_d) using the given demand function. Second, you must calculate the exact numerical elasticity at a specific price point. Finally, you are tested on the core economic meaning of elasticity: using the calculated E_d as a multiplier to determine how a percentage drop in price affects the percentage change in quantity demanded.

2. Minute Step-by-Step Solution

Given: Demand function, Q = 48 - 3P²

Step A: Find the point elasticity of demand in terms of P.
The formula for the price elasticity of demand (E_d) is:
E_d = - (P / Q) × (dQ / dP)

First, we need to find the derivative of Q with respect to P (dQ/dP).
dQ/dP = d/dP (48 - 3P²)
The derivative of a constant (48) is 0. The derivative of -3P² is -6P (using the power rule).
So, dQ/dP = -6P.
Now, substitute Q and dQ/dP into the elasticity formula:
E_d = - (P / (48 - 3P²)) × (-6P)
Multiply the numerators: -P × -6P = 6P²
E_d = 6P² / (48 - 3P²)
Trick: Simplify the fraction to make future calculations easier! Factor out a 3 from the denominator: 3(16 - P²).
E_d = 6P² / [3(16 - P²)] = 2P² / (16 - P²). (This is the answer for part a)

Step B: Find the elasticity when P = 3.
Now, substitute P = 3 into our simplified E_d formula.

E_d = 2(3)² / (16 - 3²)
E_d = 2(9) / (16 - 9)
E_d = 18 / 7
E_d ≈ 2.57 (This is the answer for part b)

Step C: Find the percentage increase in demand if the price of 3 is decreased by 4%.
By definition, Elasticity of Demand (E_d) is the ratio of the percentage change in Quantity Demanded to the percentage change in Price.

Formula: E_d = (% Change in Quantity Demanded) / (% Change in Price)
We know E_d at P = 3 is 18/7 (or 2.57).
We know the % Change in Price is 4% (a decrease, but E_d handles the inverse relationship inherently).
2.57 = (% Increase in Demand) / 4%
% Increase in Demand = 2.57 × 4%
% Increase in Demand = 10.28% (This is the answer for part c)

3. Final Tricks & Conceptual Explanation

The Core Trick: Always simplify your E_d formula algebraically before plugging in numbers (as we did by factoring out the 3 in Step A). It drastically reduces the chance of arithmetic errors during exams under time pressure.

Concept Check: Because our E_d value (2.57) is greater than 1, the demand is considered highly elastic at P = 3. This means consumers are very sensitive to price changes. A relatively small drop in price (4%) results in a proportionately massive jump in demand (10.28%). This is exactly what businesses want to calculate before deciding to run promotional discounts!

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Series: Business Mathematics - Old is Gold (Derivatives)

Question No: 2 [cite: 2010, 2012, 2013]

Topic: Application of Derivatives (Revenue Theory & Elasticity Proof)

Marks: 8 (Group C Long Answer)

Question: Suppose that the demand equation for a certain commodity is Q = 60 - 0.1P and demand elasticity is E_d.
(a) Prove the relation: MR = AR(1 - 1/E_d) where AR and MR are average and marginal revenue respectively.
(b) Use this relation to find AR when MR is 25 and MR is 2.

1. Explanation of the Question

This is a heavyweight theory and application question typically found in Group C of your exams. The first part requires you to use the Product Rule of differentiation to prove a fundamental economic theorem connecting Marginal Revenue (MR), Average Revenue (AR), and Price Elasticity of Demand (E_d). The second part is a brilliant analytical test: instead of finding MR from Q, you must manipulate the formula you just proved to work backward and find AR (which is the same as Price) given specific MR values.

2. Minute Step-by-Step Solution

Step A: Prove the relation MR = AR(1 - 1/E_d)

We know that Total Revenue (TR) is Price multiplied by Quantity:
TR = P × Q
Marginal Revenue (MR) is the derivative of TR with respect to Q. We must apply the Product Rule (uv)' = u'v + uv':
MR = d(TR)/dQ = d(P × Q)/dQ
MR = P × (dQ/dQ) + Q × (dP/dQ)
MR = P(1) + Q(dP/dQ)
Now, factor out 'P' from the expression:
MR = P [1 + (Q/P) × (dP/dQ)]
Recall two fundamental definitions:
1. Average Revenue (AR) is TR/Q. Since TR = P × Q, AR = P.
2. Point Elasticity of Demand is E_d = - (P/Q) × (dQ/dP).
Notice that the term inside our MR equation is the exact reciprocal of the elasticity formula, just without the negative sign. Therefore:
(Q/P) × (dP/dQ) = - 1 / E_d
Substitute this back into the factored MR equation:
MR = P [1 - 1/E_d]
Since P = AR, substitute AR for P:
MR = AR(1 - 1/E_d) (Proved!)

Step B: Use the relation to find AR when MR = 25 and MR = 2

Given: Demand function, Q = 60 - 0.1P

First, let's find E_d for this specific function. We know dQ/dP = -0.1.
E_d = - (P/Q) × (-0.1) = 0.1P / Q
Since we need to find AR, and we know AR = P, let's substitute 'AR' in place of 'P' everywhere. We also must replace 'Q' using the original demand equation:
Q = 60 - 0.1(AR)
Now, rewrite E_d entirely in terms of AR:
E_d = 0.1(AR) / [60 - 0.1(AR)]
Next, we need the term (1 - 1/E_d) to plug into our proven formula. Let's find 1/E_d first (just flip the fraction):
1/E_d = [60 - 0.1(AR)] / 0.1(AR)
Now calculate (1 - 1/E_d):
1 - 1/E_d = 1 - { [60 - 0.1(AR)] / 0.1(AR) }
Find a common denominator:
= [0.1(AR) - (60 - 0.1(AR))] / 0.1(AR)
= [0.2(AR) - 60] / 0.1(AR)
Now, substitute this massive term back into our proven relation MR = AR(1 - 1/E_d):
MR = AR × { [0.2(AR) - 60] / 0.1(AR) }
The 'AR' on the outside cancels with the 'AR' in the denominator! This simplifies beautifully to:
MR = [0.2(AR) - 60] / 0.1
MR = 2(AR) - 600
Now we have a direct formula linking MR and AR for this specific commodity!

Case 1: Find AR when MR = 25
25 = 2(AR) - 600
625 = 2(AR)
AR = 312.5

Case 2: Find AR when MR = 2
2 = 2(AR) - 600
602 = 2(AR)
AR = 301

3. Final Tricks & Conceptual Explanation

The Core Trick: In Part B, the trap most students fall into is calculating MR using standard derivatives (MR = dTR/dQ) and setting it to 25. While that works mathematically, the question explicitly commands you to "Use this relation [the proven formula]". By converting Q and P strictly into terms of AR, the complex elasticity fraction cancels out its own denominator, leaving you with a simple linear equation: MR = 2(AR) - 600. Always obey the specific method the question asks for to get full marks!

Concept Check: Notice that as Marginal Revenue drops from 25 to 2, the Average Revenue (Price) also drops (from 312.5 to 301). This perfectly aligns with the Law of Demand: to sell more units (which drives your marginal revenue down towards zero), a firm must lower its price.

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Series: Business Mathematics - Old is Gold (Derivatives)

Question No: 3

Topic: Application of Derivatives (Price Intervals for Elasticity)

Year Asked: 2079 (Set A, Q.No. 14)

Marks: 5

Question: The demand function for a certain commodity is given by Q = 200 - 30P. Determine the price or price intervals where the demand is:
(i) Unit elastic
(ii) Elastic
(iii) Inelastic

1. Explanation of the Question

This question moves beyond simply plugging numbers into a formula. It asks you to solve for "intervals" (ranges of numbers). In economics, demand is Unit Elastic when E_d = 1, Elastic when E_d > 1, and Inelastic when E_d < 1. Your job is to find the general formula for E_d in terms of P, and then set up inequalities to find exactly which prices cause those three conditions.

2. Minute Step-by-Step Solution

Step A: Find the general formula for Elasticity (E_d)

Given: Demand function, Q = 200 - 30P
Formula: E_d = - (P / Q) × (dQ / dP)
First, find the derivative of Q with respect to P (dQ/dP):
dQ/dP = d/dP (200 - 30P) = -30
Substitute Q and dQ/dP into the elasticity formula:
E_d = - [ P / (200 - 30P) ] × (-30)
Multiply the negatives and simplify:
E_d = 30P / (200 - 30P)

Step B: Define the economic domain (The Hidden Trick!)

In the real world, Price (P) and Quantity (Q) cannot be negative.
So, Q > 0 ⇒ 200 - 30P > 0
200 > 30P ⇒ P < 200/3 (which is approximately 66.67).
Also, P > 0. Therefore, the valid price interval for this entire problem is 0 < P < 200/3. (Keep this in mind for the inequalities!)

Step C: Find where demand is Unit Elastic

Demand is unit elastic when E_d = 1.
30P / (200 - 30P) = 1
Cross-multiply: 30P = 200 - 30P
Add 30P to both sides: 60P = 200
P = 200 / 60 = 10/3 (or approx Rs. 3.33)
Answer (i): Demand is unit elastic exactly at P = 10/3.

Step D: Find the interval where demand is Elastic

Demand is elastic when E_d > 1.
30P / (200 - 30P) > 1
Since we proved in Step B that (200 - 30P) is always positive in our valid domain, we can safely multiply both sides by it without flipping the inequality sign.
30P > 200 - 30P
60P > 200
P > 200/60 ⇒ P > 10/3
Combine this with our maximum possible price from Step B (P < 200/3).
Answer (ii): Demand is elastic in the price interval (10/3, 200/3).

Step E: Find the interval where demand is Inelastic

Demand is inelastic when E_d < 1.
30P / (200 - 30P) < 1
30P < 200 - 30P
60P < 200
P < 200/60 ⇒ P < 10/3
Combine this with our minimum possible price (P > 0).
Answer (iii): Demand is inelastic in the price interval (0, 10/3).

3. Final Tricks & Conceptual Explanation

The Core Trick: The biggest mistake students make on this question is forgetting to establish the "Domain" (Step B). If you just solve the inequality P > 10/3 and write "Demand is elastic for all prices above 10/3," you will lose marks! Why? Because if the price goes above 200/3, the quantity demanded becomes a negative number, which is economically impossible. Always cap your intervals with the natural limits of $P > 0$ and $Q > 0$.

Concept Check: This math proves a vital business concept. At very low prices (between 0 and 3.33), buyers aren't sensitive to price changes (Inelastic). But as the price gets higher and higher (between 3.33 and 66.67), buyers become highly sensitive to changes (Elastic), and raising the price further will dramatically crash your sales!

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Series: Business Mathematics - Old is Gold (Derivatives)

Question No: 4

Topic: Application of Derivatives (Profit Maximization)

Year Asked: 2074 Set C (Q.No. 48) / 2071

Marks: 5

Question: The demand function for a certain commodity is P = 20 - Q and the total cost function is C = Q² + 8Q + 2. Determine the optimal output Q, price P, total revenue R, and total profit under profit maximization.

1. Explanation of the Question

This is a fundamental "Optimization" case question. In business mathematics, maximizing profit is the ultimate goal of a firm. You are given a demand function (which dictates the Price 'P' based on Quantity 'Q') and a Cost function 'C'. To solve this, you must build your own Total Revenue (TR) and Profit (π) functions from scratch. Then, you will use the first derivative to find the critical point (where marginal profit is zero) and the second derivative to prove that this point genuinely represents a maximum peak rather than a minimum dip.

2. Minute Step-by-Step Solution

Step A: Formulate the Total Revenue (TR) Function

By definition, Total Revenue is Price multiplied by Quantity: TR = P × Q.
Substitute the given demand equation P = 20 - Q into the formula:
TR = (20 - Q) × Q
TR = 20Q - Q²

Step B: Formulate the Profit (π) Function

Profit is the money left over after subtracting Total Cost from Total Revenue: π = TR - C.
Substitute our TR formula and the given Cost (C) formula:
π = (20Q - Q²) - (Q² + 8Q + 2)
Crucial minute step: Distribute the negative sign to EVERY term in the cost function bracket!
π = 20Q - Q² - Q² - 8Q - 2
Group like terms (-Q² - Q² and 20Q - 8Q):
π = -2Q² + 12Q - 2

Step C: Apply the First-Order Condition for Maximization

To find the maximum point, we take the first derivative of the profit function with respect to Q and set it to zero (this is where the slope of the profit curve is flat).
dπ/dQ = d/dQ (-2Q² + 12Q - 2)
dπ/dQ = -4Q + 12
Set the derivative to zero: -4Q + 12 = 0
4Q = 12
Q = 3 (This is the critical quantity, but we must verify it is a maximum).

Step D: Apply the Second-Order Condition for Verification

Take the second derivative of the profit function (the derivative of -4Q + 12).
d²π/dQ² = d/dQ (-4Q + 12) = -4
Since -4 is strictly less than 0 (d²π/dQ² < 0), the function is concave downward at Q = 3. This mathematically proves that Q = 3 yields a maximum profit.
Optimal Output (Q) = 3 units.

Step E: Calculate the required final values at Q = 3

Optimal Price (P): Substitute Q = 3 into the original demand function.
P = 20 - Q ⇒ P = 20 - 3 ⇒ P = Rs. 17
Total Revenue (TR): Substitute Q = 3 into our TR function.
TR = 20Q - Q² ⇒ 20(3) - (3)² ⇒ 60 - 9 ⇒ TR = Rs. 51
Maximum Profit (π): Substitute Q = 3 into our simplified Profit function.
π = -2(3)² + 12(3) - 2 ⇒ -2(9) + 36 - 2 ⇒ -18 + 36 - 2 ⇒ Maximum Profit = Rs. 16

3. Final Tricks & Conceptual Explanation

The Core Trick: The most common error students make here is the "Sign Drop Trap" in Step B. When subtracting the cost function, students often write - Q² + 8Q + 2 instead of enclosing it in brackets - (Q² + 8Q + 2). If you forget to flip the signs of the +8Q and +2, your entire profit function will be wrong, leading to a cascade of incorrect answers. Always use brackets when subtracting a polynomial!

Concept Check: What does the math actually mean? It means if the factory produces only 1 or 2 units, they aren't covering their fixed and variable costs optimally. If they produce 4 or more units, the market price drops too low (due to P = 20 - Q) and their marginal cost of production starts eating into their margins. Exactly 3 units at Rs. 17 is the "sweet spot" where Marginal Revenue perfectly equals Marginal Cost, leaving them with the absolute highest possible profit of Rs. 16.

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Series: Business Mathematics - Old is Gold (Integration)

Question No: 5

Topic: Application of Definite Integrals (Consumer & Producer Surplus)

Year Asked: Frequently Repeated (Group C Long Question)

Marks: 8

Question: The demand and supply curves of an item are given by the equations $P_d = 20 - 3Q - Q^2$ and $P_s = Q - 1$ respectively. Find the difference between Consumer Surplus (CS) and Producer Surplus (PS) at the equilibrium price.

1. Explanation of the Question

We are now transitioning from Derivatives to Integration! This is a massive 8-mark question from Group C that combines market economics with definite integrals. To solve it, you first need to find the "Equilibrium Point"—the exact price and quantity where the demand curve intersects the supply curve. Once you have that, you must use definite integration to calculate the area under the demand curve (Consumer Surplus) and the area above the supply curve (Producer Surplus), and finally find the difference between the two.

2. Minute Step-by-Step Solution

Step A: Find the Market Equilibrium

At market equilibrium, Quantity Demanded equals Quantity Supplied, which means their prices must be equal: $P_d = P_s$.
Substitute the given equations:
$$20 - 3Q - Q^2 = Q - 1$$
Bring all terms to one side to form a standard quadratic equation ($aQ^2 + bQ + c = 0$):
$$0 = Q^2 + Q + 3Q - 20 - 1$$
$$Q^2 + 4Q - 21 = 0$$
Now, factorize the quadratic equation. We need two numbers that multiply to -21 and add to 4 (which are 7 and -3):
$$Q^2 + 7Q - 3Q - 21 = 0$$
$$Q(Q + 7) - 3(Q + 7) = 0$$
$$(Q - 3)(Q + 7) = 0$$
This gives $Q = 3$ or $Q = -7$. Since quantity cannot be negative in the real world, we reject -7.
Equilibrium Quantity ($Q_e$) = 3.
Substitute $Q_e = 3$ into either the supply or demand equation to find the Equilibrium Price ($P_e$). Using the simpler supply equation:
$P_e = Q_e - 1 = 3 - 1$
Equilibrium Price ($P_e$) = 2.

Step B: Calculate Consumer Surplus (CS)

The formula for Consumer Surplus is the definite integral of the demand function from $0$ to $Q_e$, minus the total actual revenue ($P_e \times Q_e$):
$$CS = \int_{0}^{Q_e} P_d \, dQ - (P_e \times Q_e)$$
Plug in our values ($Q_e = 3$, $P_e = 2$):
$$CS = \int_{0}^{3} (20 - 3Q - Q^2) \, dQ - (2 \times 3)$$
Integrate the function using the power rule ($\int Q^n dQ = \frac{Q^{n+1}}{n+1}$):
$$CS = \left[ 20Q - \frac{3Q^2}{2} - \frac{Q^3}{3} \right]_{0}^{3} - 6$$
Apply the limits (plug in 3, the 0 limit just becomes 0):
$$CS = \left( 20(3) - \frac{3(3)^2}{2} - \frac{(3)^3}{3} \right) - 6$$
$$CS = \left( 60 - \frac{27}{2} - \frac{27}{3} \right) - 6$$
$$CS = (60 - 13.5 - 9) - 6$$
$$CS = 37.5 - 6 = 31.5$$
Consumer Surplus (CS) = 31.5

Step C: Calculate Producer Surplus (PS)

The formula for Producer Surplus is the total actual revenue ($P_e \times Q_e$) minus the definite integral of the supply function from $0$ to $Q_e$:
$$PS = (P_e \times Q_e) - \int_{0}^{Q_e} P_s \, dQ$$
Plug in our values:
$$PS = (2 \times 3) - \int_{0}^{3} (Q - 1) \, dQ$$
Integrate the supply function:
$$PS = 6 - \left[ \frac{Q^2}{2} - Q \right]_{0}^{3}$$
Apply the limits:
$$PS = 6 - \left( \frac{(3)^2}{2} - 3 \right)$$
$$PS = 6 - (4.5 - 3)$$
$$PS = 6 - 1.5 = 4.5$$
Producer Surplus (PS) = 4.5

Step D: Find the Difference

The question asks for the difference between Consumer and Producer Surplus.
Difference = CS - PS
Difference = 31.5 - 4.5
Final Answer = 27

3. Final Tricks & Conceptual Explanation

The Core Trick: The biggest mistake students make in Integration applications is forgetting the $-(P_e \times Q_e)$ in the CS formula, or swapping the order in the PS formula. A trick to remember: Consumers want prices to drop, so their curve is high, meaning the integral comes first (it's the bigger area). Producers want prices to rise, so their baseline is the $P_e \times Q_e$ box, making the integral come second.

Concept Check: What do these numbers mean? The Consumer Surplus of 31.5 represents the extra "benefit" or savings that buyers got because they were willing to pay much more than the equilibrium price of Rs. 2, but didn't have to. The Producer Surplus of 4.5 represents the extra profit sellers made because they were willing to sell the items for less than Rs. 2, but the market allowed them to charge Rs. 2. The difference of 27 is a comparative measure of market welfare favoring the consumers in this specific scenario.

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Series: Business Mathematics - Old is Gold (Probability)

Question No: 1 (Subjective Questions)

Topic: Conditional Probability & Bayes' Theorem

Year Asked: 2082 (Q.No. 16)

Marks: 5

Question: In a cap factory, machines A, B, and C manufacture respectively 30%, 35%, and 35% of the total output. Out of their output, 2%, 3%, and 4% are defective caps respectively. A cap is drawn at random and is found to be defective. What is the probability that it was manufactured by machine A?

1. Explanation of the Question

This is a classic Bayes' Theorem problem. It deals with "reverse probability." Normally, you go forward: you pick a machine, then find a defective cap. Here, you are going backward: you already know the cap is defective (this is your given condition), and you need to trace back to find the probability that it originally came from a specific source (Machine A). To solve this, we will first find the total probability of getting a defective cap from anywhere in the factory, and then find what fraction of that total defectiveness is specifically Machine A's fault.

2. Minute Step-by-Step Solution

Step A: Define the Events

Let $E_1$ = Event that the cap is manufactured by Machine A.
Let $E_2$ = Event that the cap is manufactured by Machine B.
Let $E_3$ = Event that the cap is manufactured by Machine C.
Let $D$ = Event that the drawn cap is Defective.

Step B: List the Given Probabilities (Prior Probabilities)

Probability of choosing Machine A: $P(E_1) = 30\% = 0.30$
Probability of choosing Machine B: $P(E_2) = 35\% = 0.35$
Probability of choosing Machine C: $P(E_3) = 35\% = 0.35$
Self-check: $0.30 + 0.35 + 0.35 = 1.00$. This confirms we have covered 100% of the factory's output.

Step C: List the Conditional Probabilities (Likelihoods)

Probability of a defect given it's from Machine A: $P(D|E_1) = 2\% = 0.02$
Probability of a defect given it's from Machine B: $P(D|E_2) = 3\% = 0.03$
Probability of a defect given it's from Machine C: $P(D|E_3) = 4\% = 0.04$

Step D: State the Formula (Bayes' Theorem)

We want to find the probability that it came from Machine A, given that it is defective: $P(E_1|D)$.

$$P(E_1|D) = \frac{P(E_1) \cdot P(D|E_1)}{P(E_1) \cdot P(D|E_1) + P(E_2) \cdot P(D|E_2) + P(E_3) \cdot P(D|E_3)}$$

Step E: Calculate the Denominator (Total Probability of Defect)

Defects from A: $0.30 \times 0.02 = 0.006$
Defects from B: $0.35 \times 0.03 = 0.0105$
Defects from C: $0.35 \times 0.04 = 0.014$
Total $P(D)$ = $0.006 + 0.0105 + 0.014 = 0.0305$ (So, 3.05% of all caps in the factory are defective).

Step F: Final Calculation

Now, plug the numerator (Defects from A) and the denominator (Total Defects) into the formula:
$$P(E_1|D) = \frac{0.006}{0.0305}$$
To simplify, multiply top and bottom by 10,000 to remove decimals:
$$P(E_1|D) = \frac{60}{305}$$
Divide top and bottom by 5:
$P(E_1|D) = \frac{12}{61}$ (or approximately 0.1967 / 19.67%)

3. Hints, Tips & Conceptual Explanation

The Core Trick: When memorizing Bayes' Theorem, think of it in words rather than just variables: Target Path / Sum of All Paths. The numerator is just the specific branch of the probability tree you care about (Machine A making a defect). The denominator is every possible way a defect could happen added together. If you calculate the bottom (Total Probability) first, you just put your target number on top of it!

Common Mistake Avoidance: Always convert your percentages to decimals ($3\% = 0.03$, NOT $0.3$). Writing $0.3$ for $3\%$ is the most common reason students lose marks on this question. Also, always verify that your base machine probabilities ($P(E_1) + P(E_2) + P(E_3)$) add exactly to 1.00.

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Series: Business Mathematics - Old is Gold (Probability - Long Questions)

Question No: 8 (From Subjective Questions Section)

Topic: Probability (Binomial Distribution)

Year Asked: 2081 (Q.No. 18)

Marks: 5 (1 + 2 + 2)

Question: In a college, it has been found that 40% of the students withdraw without completing the mathematics course. If there were 10 students have registered for the course of this semester.
a. What is the probability that none will withdraw? [1]
b. What is the probability that at least one will withdraw? [2]
c. What is the probability at most one will withdraw? [2]

1. Explanation of the Question

This is a comprehensive, 5-mark long question based on the Binomial Distribution. It tests your ability to identify the parameters ($n$, $p$, and $q$) from a word problem and apply the binomial formula to different scenarios.

In mathematics, a "success" doesn't always mean something good; it just means the event we are tracking. Here, we are tracking students who withdraw. So, a student withdrawing is our mathematical "success" ($p = 40\%$). We have a fixed number of trials (10 students), making it a perfect binomial scenario. You will need to calculate probabilities for exactly zero, one or more, and zero or one.

2. Minute Step-by-Step Solution

Step A: Identify the Binomial Parameters

Let $X$ be the random variable representing the number of students who withdraw.
Total number of students (trials), $n = 10$
Probability of a student withdrawing (success), $p = 40\% = 0.40$
Probability of a student NOT withdrawing (failure), $q = 1 - p = 1 - 0.40 = 0.60$

Step B: State the Binomial Formula

The probability of exactly $x$ students withdrawing is given by:
$$P(X = x) = \binom{n}{x} \cdot p^x \cdot q^{n-x}$$

Step C: Solve Part (a) - Probability that NONE will withdraw

"None" means $x = 0$.
$P(X = 0) = \binom{10}{0} \cdot (0.40)^0 \cdot (0.60)^{10}$
We know that $\binom{10}{0} = 1$ and anything to the power of 0 is $1$.
$P(X = 0) = 1 \cdot 1 \cdot (0.60)^{10}$
$P(X = 0) \approx 0.006046$
Answer (a): The probability that none will withdraw is approximately 0.0060 (or 0.60%).

Step D: Solve Part (b) - Probability that AT LEAST ONE will withdraw

"At least one" means $X \ge 1$ (which includes 1, 2, 3... all the way to 10).
Instead of calculating 10 different probabilities and adding them, use the complement rule: the opposite of "at least one" is "none".
$P(X \ge 1) = 1 - P(X = 0)$
We already found $P(X = 0)$ in Step C!
$P(X \ge 1) = 1 - 0.006046$
$P(X \ge 1) = 0.993954$
Answer (b): The probability that at least one will withdraw is approximately 0.9940 (or 99.40%).

Step E: Solve Part (c) - Probability that AT MOST ONE will withdraw

"At most one" means a maximum of 1. This includes exactly 0 OR exactly 1. So, $X \le 1$.
$P(X \le 1) = P(X = 0) + P(X = 1)$
We already have $P(X = 0) = 0.006046$. Now we need $P(X = 1)$.
$P(X = 1) = \binom{10}{1} \cdot (0.40)^1 \cdot (0.60)^9$
$P(X = 1) = 10 \cdot 0.40 \cdot (0.0100776)$
$P(X = 1) = 4 \cdot 0.0100776 \approx 0.04031$
Now, add them together:
$P(X \le 1) = 0.006046 + 0.040310$
$P(X \le 1) = 0.046356$
Answer (c): The probability that at most one will withdraw is approximately 0.0464 (or 4.64%).

3. Final Hints & Conceptual Tips

The "Success" Trap: In everyday language, passing a course is a "success." However, in probability modeling, "success" is strictly defined as the event you are trying to calculate. Because the question asks for the probability of students withdrawing, withdrawing becomes our $p$ (0.40). If you accidentally make $p = 0.60$ (students who stay), every single answer will be backward!

Calculator Tip for Powers: When dealing with high powers like $(0.60)^{10}$, keep at least 5 or 6 decimal places during your intermediate steps. If you round off too early (e.g., rounding $(0.60)^9$ to $0.01$), your final addition in part (c) will be significantly off, which can cost you marks on accuracy.