Problem 3 - Algebra for JEE ADVANCED

Question

A forecast is to be made of the results of five cricket matches, each of which can be a win or a draw or a loss for Indian team. Let, \(p=\) number of forecasts with exactly I error \(q=\) number of forecasts with exactly 3 errors and \(r=\) number of forecasts with all five errors Then the correct statement(s) is/are a. \(2 q=5 r\) b. \(8 p=q\) c. \(8 p=5 r\) d. \(2(p+r)>q\)

Step-by-Step Solution

Answer

Correct statements are a, b, and d.

Step 1: Total Forecasts and Errors

Each match can end in three different outcomes: win, draw, or loss. Therefore, the number of possible outcomes for 5 matches is \(3^5 = 243\). A forecast involves predicting the win, draw, or loss for these 5 matches. Thus, there are 243 total forecasts possible.

Step 2: Calculating Exact Error Forecasts

We need to calculate the number of forecasts with exactly 1 error, 3 errors, and all 5 errors. Using the combination formula and considering one outcome can be incorrect:- For exactly 1 error: \( \binom{5}{1} \times 2^1 = 5 \times 2 = 10 \). So, \( p = 10 \).- For exactly 3 errors: \( \binom{5}{3} \times 2^3 = 10 \times 8 = 80 \). So, \( q = 80 \).- For exactly 5 errors: \( \binom{5}{5} \times 2^5 = 1 \times 32 = 32 \). So, \( r = 32 \).

Step 3: Analyze Statements

Evaluate each statement using \(p = 10\), \(q = 80\), and \(r = 32\):- **Statement (a)**: \(2q = 5r\) becomes \(160 = 160\), which is true.- **Statement (b)**: \(8p = q\) becomes \(80 = 80\), which is true.- **Statement (c)**: \(8p = 5r\) becomes \(80 = 160\), which is false.- **Statement (d)**: \(2(p + r) > q\) becomes \(2(10 + 32) = 84 > 80\), which is true.

Key Concepts

Permutation and Combination

In combinatorics, permutation and combination are essential concepts that help in calculating the number of ways objects can be arranged or selected. Permutations consider the order of arrangement, while combinations focus on selecting objects without regard to order.
For example, in the context of choosing outcomes for 5 cricket matches, the order of wins, draws, and losses matters. However, when counting forecasts with errors, we're mainly interested in the number of incorrect selections, which draws on the concept of combinations.

• Permutations: Useful when the sequence or arrangement is important.
• Combinations: Helpful for counting selections where order doesn't matter, like in our exercise problem, where selecting matches for errors employs combinations.

Combinatorial calculations often include expressions like \( \binom{n}{k} \), which denotes the number of ways to choose \( k \) items from \( n \) total items. In this exercise, combinations are computed for exactly 1 error, 3 errors, and 5 errors, using the binomial coefficient symbolically represented as \( \binom{5}{1} \), \( \binom{5}{3} \), and \( \binom{5}{5} \). These calculations help determine how many forecasts have a certain number of errors.

Error Analysis

Error analysis in forecasting involves understanding the inaccuracies in predictions. It is crucial for improving prediction models. In our exercise, error analysis focuses on identifying how many forecasts have a specific number of errors when predicting the outcomes of cricket matches.
The exercise uses mathematical combinations to calculate forecasts with specific errors. Here's how it breaks down:

• Exactly 1 error: You choose 1 match out of 5 to be wrong, with the other 4 being right. Hence, combinations \( \binom{5}{1} = 5 \) and 2 choices of alternative outcomes.
• Exactly 3 errors: Select 3 matches to be wrong. So, \( \binom{5}{3} = 10 \) with 2 alternative choices for each wrong outcome.
• All 5 errors: All guesses are wrong, combined as \( \binom{5}{5} = 1 \) with the wrong option having 2 alternative outcomes.

Understanding how these errors distribute amongst forecasts helps refine models to make better predictions by minimizing future errors.

Probability in Sports Forecasting

The use of probability in sports forecasting involves predicting the likely outcomes of sporting events. It combines statistical analysis, historical data, and other influencing factors to assess how likely different results are to occur.
In this exercise, you are essentially making sports forecasts for cricket matches by looking at potential outcomes and the likelihood of making errors. Calculating the probability of each outcome is crucial for improving the accuracy of predictions. With each cricket match having the possibility of three distinct results, the total forecast number is calculated using a power of three due to three possible outcomes per match.

• Analyzing historical sports data can give insights into probable outcomes.
• Understanding probability distributions can help in creating accurate forecasts.
• Evaluating forecast errors as seen in the exercise helps recognize where predictions might often go wrong.

By applying probability, sports analysts can quantify expectations for win, draw, or loss, and better refine their forecasting models to aid teams, bettors, and fans.