Probability Calculation Winning First And Second Prize With Mom
In this article, we will delve into a fascinating probability problem involving a drawing with multiple prizes. Let's imagine a scenario where you and your mom decide to participate in a drawing that offers three distinct prizes. A total of nine individuals have entered this drawing, and the prizes are to be awarded randomly. The question at hand is to determine the probability of a specific outcome: you winning the first prize and your mom winning the second prize. To solve this, we'll explore the concepts of permutations, combinations, and probability, providing a comprehensive explanation to make the solution clear and understandable.
Understanding the Problem
Before we dive into the calculations, let's make sure we fully understand the problem. We know that there are three different prizes up for grabs and a total of nine participants. The crucial aspect here is that the prizes are distinct, meaning that winning first prize is different from winning second or third prize. This distinction is important because it affects how we calculate the number of possible outcomes. We need to figure out the likelihood of you securing the top spot and your mom taking the second prize, a specific arrangement out of all the possible ways the prizes could be awarded. The problem states that there are 504 ways to award the prizes, a piece of information that we will use to calculate the desired probability.
Calculating the Total Number of Ways to Award the Prizes
The first step in solving this probability problem is to determine the total number of ways the prizes can be awarded. Since the prizes are distinct, the order in which they are awarded matters. This means we are dealing with a permutation problem. A permutation is an arrangement of objects in a specific order. The number of permutations of n objects taken r at a time is denoted as P(n, r) and is calculated using the formula:
P(n, r) = n! / (n - r)!
Where "!" denotes the factorial function (e.g., 5! = 5 × 4 × 3 × 2 × 1). In our case, we have 9 participants (n = 9) and 3 prizes (r = 3). So, we need to calculate P(9, 3):
P(9, 3) = 9! / (9 - 3)! P(9, 3) = 9! / 6! P(9, 3) = (9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1) / (6 × 5 × 4 × 3 × 2 × 1) P(9, 3) = 9 × 8 × 7 P(9, 3) = 504
This calculation confirms the information given in the problem: there are indeed 504 ways to award the three prizes among the nine participants. Understanding this total number of possibilities is crucial for calculating the probability of our specific event.
Determining the Number of Favorable Outcomes
Now that we know the total number of ways to award the prizes, we need to determine the number of outcomes where you win first prize and your mom wins second prize. This is our favorable outcome. If you win first prize, there is only one possibility for that outcome. Similarly, if your mom wins second prize, there is also only one possibility for that outcome. After you and your mom have been awarded the first and second prizes, there is only one prize left to be awarded, and there are 7 remaining participants who could win it.
So, to find the number of favorable outcomes, we need to consider the following:
- You win first prize: 1 possibility
- Your mom wins second prize: 1 possibility
- One of the remaining 7 people wins the third prize: 7 possibilities
To get the total number of favorable outcomes, we multiply these possibilities together:
1 × 1 × 7 = 7
Therefore, there are 7 favorable outcomes where you win first prize and your mom wins second prize.
Calculating the Probability
Now that we know the total number of possible outcomes (504) and the number of favorable outcomes (7), we can calculate the probability of you winning first prize and your mom winning second prize. Probability is defined as the number of favorable outcomes divided by the total number of possible outcomes:
Probability = (Number of Favorable Outcomes) / (Total Number of Possible Outcomes)
In our case:
Probability = 7 / 504
We can simplify this fraction by dividing both the numerator and the denominator by 7:
Probability = 1 / 72
So, the probability of you winning first prize and your mom winning second prize is 1/72.
Expressing the Probability
We have calculated the probability as a fraction (1/72), but it can also be expressed as a decimal or a percentage. To convert the fraction to a decimal, we simply divide the numerator by the denominator:
1 / 72 ≈ 0.0139
To express this decimal as a percentage, we multiply by 100:
- 0139 × 100 ≈ 1.39%
Therefore, the probability of you winning first prize and your mom winning second prize is approximately 0.0139 or 1.39%. This means that there is a relatively small chance of this specific outcome occurring.
Conclusion
In this article, we have solved a probability problem involving a drawing with three distinct prizes and nine participants. We calculated the probability of you winning first prize and your mom winning second prize by first determining the total number of possible outcomes (504) using permutations. Then, we calculated the number of favorable outcomes (7) where you win first prize and your mom wins second prize. Finally, we divided the number of favorable outcomes by the total number of possible outcomes to arrive at the probability of 1/72, or approximately 1.39%. This exercise demonstrates the application of probability principles in a real-world scenario. Understanding these concepts allows us to make informed decisions and assess the likelihood of various events occurring.
Understanding probability can be incredibly useful in many aspects of life, from understanding the odds in games of chance to making informed decisions about investments and risk assessment. By breaking down complex problems into smaller, manageable steps, we can gain a clearer understanding of the world around us and make better choices.
This problem highlights the importance of carefully considering all possible outcomes and the specific conditions of a situation when calculating probabilities. The distinction between permutations and combinations, for example, is crucial, as using the wrong approach can lead to incorrect results. Similarly, understanding the concept of favorable outcomes and how to identify them is essential for accurate probability calculations. By mastering these principles, you can tackle a wide range of probability problems and develop a stronger understanding of statistical reasoning.
In conclusion, the probability of you winning first prize and your mom winning second prize in this drawing is 1/72, or approximately 1.39%. While this is a relatively small probability, it is a testament to the random nature of the drawing and the possibility of any participant winning. This exercise serves as a valuable example of how probability calculations can be used to quantify the likelihood of specific events and gain a better understanding of the world around us.
