You and your mom enter a drawing with 3 different prizes. A total of 9 people entered the drawing, and prizes are awarded randomly.
There are 504 ways to award the prizes.
What is the probability that you win first prize and your mom wins second prize?
A. [tex]$\frac{8}{504}$[/tex]
B. [tex]$\frac{1}{9}$[/tex]
C. [tex]$\frac{7}{504}$[/tex]
D. [tex]$\frac{1}{504}$[/tex]
Answer (2)
Determine the number of ways you win first and your mom wins second: 7.
Divide the number of ways you and your mom win by the total number of ways to award prizes: 504 7 .
Simplify the fraction: 504 7 = 72 1 .
The probability that you win first prize and your mom wins second prize is: 504 7 .
Explanation
Understanding the Problem Let's break down this probability problem step by step! We're trying to find the chance that you win first prize and your mom wins second prize in a drawing with 3 prizes and 9 total participants. We know there are 504 total possible ways the prizes could be awarded.
Determining Favorable Outcomes First, let's figure out how many ways you can win first prize and your mom wins second prize. If you have first prize and your mom has second prize, that leaves one prize to be awarded. There are 7 people remaining (since you and your mom already have prizes), so there are 7 possible people who could win the third prize.
Calculating the Probability Now, to find the probability, we'll divide the number of favorable outcomes (you win first, mom wins second) by the total number of possible outcomes. We found that there are 7 favorable outcomes, and we were given that there are 504 total possible outcomes. So, the probability is 504 7 .
Simplifying the Fraction We can simplify the fraction 504 7 by dividing both the numerator and the denominator by 7. This gives us 72 1 . Now we need to compare this result with the given options. We can also represent this probability as a decimal: 504 7 ≈ 0.0139 .
Final Answer Comparing our simplified probability 504 7 with the answer choices, we see that option C, 504 7 , matches our calculated probability. Therefore, the probability that you win first prize and your mom wins second prize is 504 7 .
Examples
Imagine you're organizing a raffle to raise money for a local charity. Understanding probability helps you determine the fairness of the raffle and the likelihood of specific people winning. For instance, if you want to ensure that a particular group has a higher chance of winning, you can adjust the number of tickets they receive. This ensures a transparent and equitable process, boosting trust and participation in your fundraising efforts. The probability of you and your mom winning the first and second prize can be calculated as 504 7 .
The probability that you win first prize and your mom wins second prize is calculated by dividing the number of favorable outcomes (7) by the total outcomes (504), resulting in 504 7 . This is the answer that corresponds to option C. The exact probability is not simplified to match other options but should be realized as part of the broader understanding of probability outcomes.
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