Problem 21 - Algebra 2 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

India plays two matches each with West Indies and Australia. In any match the probabilities of India getting point 0,1 and 2 are \(0.45,0.05\) and \(0.50\), respectively. Assuming that the outcomes are independents, the probability of India getting at least 7 points is (a) \(0.8750\) (b) \(0.0875\) (c) \(0.0625\) (d) \(0.0250\)

Step-by-Step Solution

Answer

The probability of India getting at least 7 points is 0.0875.

Step 1: Understand the Situation

India plays a total of 4 matches, two each with West Indies and Australia. In each match, India can earn 0, 1, or 2 points with probabilities 0.45, 0.05, and 0.50, respectively. We need to find the probability that India earns at least 7 points in total from these matches.

Step 2: Determine Possible Outcomes

The total number of points India can earn ranges from a minimum of 0 (if India scores 0 points in all 4 matches) to a maximum of 8 points (if India scores 2 points in each match). We are interested in cases where the total points are at least 7, meaning India can achieve either 7 or 8 points.

Step 3: Calculate Probability for 7 Points

To score exactly 7 points, India might achieve several combinations. One possible way is scoring three 2-point games and one 1-point game. The probability for this outcome is calculated using \( P(2,2,2,1) \). The combination is \( \binom{4}{3} \), representing choosing 3 matches out of 4 to score 2 points.

Step 4: Specific Calculation for 7 Points

\[ P(7 \, \text{points}) = \binom{4}{3} \times (0.5)^3 \times (0.05)^1 = 4 \times 0.125 \times 0.05 = 0.025 \]

Step 5: Calculate Probability for 8 Points

To score exactly 8 points, India must score 2 points in all four matches. So the probability is simply \( (0.5)^4 \).

Step 6: Specific Calculation for 8 Points

\[ P(8 \, \text{points}) = (0.5)^4 = 0.0625 \]

Step 7: Add Probabilities of 7 and 8 Points

The total probability of India scoring at least 7 points (either 7 or 8 points) is the sum of the individual probabilities: \( P(7 \, \text{points}) + P(8 \, \text{points}) \).

Step 8: Final Probability Calculation

\[ P(\text{at least 7 points}) = 0.025 + 0.0625 = 0.0875 \]

Key Concepts

Probability Calculation

Probability calculation helps us predict the chance of certain events occurring. In games and real-life situations, this is vital to make informed decisions. Let's break down how you can calculate probability in the context of India playing cricket matches.
Each match shows different outcomes: India scoring 0, 1, or 2 points. Understanding the probability of these outcomes is the first step:

• The probability of India scoring 0 points is 0.45.
• The probability of 1 point is 0.05.
• The probability of 2 points is 0.50.

To find the probability of a certain total score across several matches, you sum the probabilities of individual game outcomes. In this exercise, we're interested in at least 7 points in 4 matches. Focus on the scenarios where India scores 7 or 8 points, and sum those probabilities for the final answer.

Combinatorics in Probability

Combinatorics is a tool used to count combinations, and it's crucial in probability for calculating complex outcomes. In our exercise, combinatorics help identify the number of ways India can score 7 points in 4 matches.
To score 7 points, India could achieve it by scoring three 2-point games and one 1-point game. Calculating for this combination:

• First, choose 3 matches where India scores 2 points. This can be done using the combination formula: \( \binom{4}{3} \). This measures how many ways we can choose 3 matches out of 4.
• Multiply this arrangement by the probabilities: \( (0.5)^3 \times (0.05)^1 \).
• The calculation looks like: \( P(7) = \binom{4}{3} \times 0.5^3 \times 0.05 = 4 \times 0.125 \times 0.05 = 0.025 \).

Combinatorics allow us to structure and solve probabilities accurately by assessing all possible combinations.

Independent Events

Independence of events is a crucial concept in probability. Two events are independent if the occurrence of one does not influence the other. In this exercise, each cricket match is an independent event because the outcome of one match doesn't affect the others.
When events are independent, we calculate the probability of combined outcomes by multiplying the probabilities of each independent event.

• For example, to find the probability of India scoring 8 points (all 2-point games), we calculate the probability of 2 points in each of the 4 matches: \( P(8) = (0.5)^4 = 0.0625 \).
• This multiplication is possible only because individual match outcomes are independent.

Understanding independence helps us accurately calculate the probability of complex occurrences across multiple events, like sports matches, by simplifying the calculations to multiplication of their individual outcomes.