Problem 11 - Algebra 2 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

\(A\) has 3 shares in a lottery containing 3 prizes and 9 blanks. \(B\) has two shares in a lottery containing 2 prizes and 6 blanks; Find theratio of \(A\) 's chance of success to \(B\) 's chance of success. (a) \(927: 715\) (b) \(972: 751\) (c) \(925: 715\) (d) \(715: 972\) olution (c) Since \(A\) has 3 shares in a lottery, his chance of success means that he gets at least 1 prize, that is, he gets either 1 prize or 2 prizes or 3 prizes and his chance of failure means that he gets no prize. It is certain that either he succeeds or fails. If \(p\) denotes his chance of success and \(q\) the chance of his failure, then \(p+q=1\) or \(p=1-q\) We now find \(q \times n=\) total number of ways $$ ={ }^{12} C_{3}=\frac{12 \times 11 \times 10}{1 \times 2 \times 3}=220 $$ Since out of 12 tickets in the lottery, he can draw any 3 tickets by virtue of his having 3 shares in the lottery and \(m=\) favourable number of ways $$ ={ }^{9} C_{3}=\frac{9 \times 8 \times 7}{1 \times 2 \times 3}=84 $$ Since he will fail to draw a prize if all the tickets drawn by him are blanks. $$ \therefore q=\frac{m}{n}=\frac{84}{220}=\frac{21}{55} $$ \(\therefore p=A\) 's chance of success \(=1-\frac{21}{55}=\frac{34}{55}\)Similarly B's chance of success $$ \begin{aligned} p^{\prime} &=1-q^{\prime}=1-\frac{{ }^{6} C_{2}}{{ }^{8} C_{2}}=1-\frac{6 \times 5}{8 \times 7} \\ &=1-\frac{15}{28}=\frac{13}{28} \end{aligned} $$ \(\therefore A\) 's chance of success: B's chance of success $$ =p: p^{\prime}=\frac{34}{35}: \frac{13}{28}=\frac{952}{1540}: \frac{715}{1540}=952: 715 $$

Step-by-Step Solution

Answer

The ratio of A's chance of success to B's is \(952:715\) (option c).

Step 1 - Calculate total outcomes for A

A has 3 shares from 12 tickets (3 prizes + 9 blanks), so calculate the number of ways to draw 3 tickets: \[ n = {^{12}C_3} = \frac{12 \times 11 \times 10}{1 \times 2 \times 3} = 220 \]

Step 2 - Calculate unsuccessful outcomes for A

Calculate the number of ways A can draw 3 blanks (failure): \[ m = {^{9}C_3} = \frac{9 \times 8 \times 7}{1 \times 2 \times 3} = 84 \]

Step 3 - Calculate A's chance of failure

A's chance of failure is the ratio of unsuccessful outcomes to total outcomes: \[ q = \frac{m}{n} = \frac{84}{220} = \frac{21}{55} \]

Step 4 - Calculate A's chance of success

Since A's chance of success, \( p = 1 - q \): \[ p = 1 - \frac{21}{55} = \frac{34}{55} \]

Step 5 - Calculate total outcomes for B

B has 2 shares from 8 tickets (2 prizes + 6 blanks), so calculate the number of ways to draw 2 tickets: \[ n' = {^{8}C_2} = \frac{8 \times 7}{1 \times 2} = 28 \]

Step 6 - Calculate unsuccessful outcomes for B

Calculate the number of ways B can draw 2 blanks (failure): \[ m' = {^{6}C_2} = \frac{6 \times 5}{1 \times 2} = 15 \]

Step 7 - Calculate B's chance of failure

B's chance of failure is the ratio of unsuccessful outcomes to total outcomes: \[ q' = \frac{m'}{n'} = \frac{15}{28} \]

Step 8 - Calculate B's chance of success

Since B's chance of success, \( p' = 1 - q' \): \[ p' = 1 - \frac{15}{28} = \frac{13}{28} \]

Step 9 - Calculate the ratio of success chances

Calculate the ratio of A's chance of success to B's chance of success: \[ \text{Ratio} = \frac{p}{p'} = \frac{\frac{34}{55}}{\frac{13}{28}} = \frac{34 \times 28}{55 \times 13} = \frac{952}{715} \]

Key Concepts

Combinatorics

Combinatorics is a fundamental concept in probability that involves counting, arranging, and choosing the objects in a particular context. It helps us calculate the number of possible outcomes or combinations in scenarios where we have a variety of objects or choices. This principle is applied in the lottery problem to understand the chance of drawing specific types of tickets. Combinatorics uses mathematical tools like permutations and combinations to efficiently handle counting problems.

In this specific lottery context, combinations are used to determine the number of ways tickets can be chosen. The notation \({^n}C_r\) represents the number of combinations where \( n \) is the total number of items to choose from, and \( r \) is the number of items to choose. For instance, to calculate \\( {^{12}C_3} \,\) in the problem, we find the number of ways 3 tickets can be drawn from the 12 available tickets. This is depicted by the formula:

• \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \)

where \(!\) indicates factorial, meaning the product of all positive integers up to that number. Thus, combinatorics provides systematic methods to compute these possibilities, which are crucial in assessing probabilities in complex scenarios.

Ratio and Proportion

Ratio and proportion are mathematical concepts that describe relationships or comparisons between numbers. In probability, these concepts are used to compare the likelihood of different outcomes.

In this exercise, we are calculating the ratio of the chances of success for two lottery players, A and B. To find this ratio, we need to first calculate their individual chances of success using the concept of probability:

• The probability of an event happening is calculated as the ratio of the number of favorable outcomes to the total number of possible outcomes.

For player A, we determined that the chance of them winning (success) involves them drawing at least one prize ticket. By calculating combinations for drawing blanks, we can determine their probability of failure and hence their chance of success, \( p = 1 - q \). Similarly, we compute for player B. Then, their ratios are compared as follows:

This accurately provides a ratio, 952:715, portraying how A's chance of winning compares to B's, which gives insight into their relative probabilities.

Lottery Problems

Lottery problems are a classic exercise in understanding probability and chance, often framed as determining the likelihood of success in a scenario with various outcomes. These problems require a clear understanding of how probabilities work and rely heavily on combinatorial calculations. In a typical lottery scenario, tickets can be either prizes or blanks. Participants must determine the odds of drawing a winning ticket.

For example, consider player A who participates with multiple lottery shares. Their task is to calculate the probability of drawing at least one prize from an array of prize and blank tickets. This involves:

• Understanding the total number of possible outcomes (all combinations of tickets),
• Calculating the number of favorable outcomes (drawing a prize),
• Finding the ratio of favorable outcomes to total possible outcomes.

Lottery problems help illustrate practical applications of probability principles and are excellent exercises for understanding how chance operates in everyday situations. These problems also highlight the crucial mathematical techniques necessary in assessing the viability of participation in chance-driven activities.