Thank you for visiting To practice for a Colorado Spelling Bee Contest Arapahoe Ridge High A and Boulder High B have selected their best five spellers to compete at. This page is designed to guide you through key points and clear explanations related to the topic at hand. We aim to make your learning experience smooth, insightful, and informative. Dive in and discover the answers you're looking for!
Answer :
Final answer:
The probability mass function (PMF) of H, which represents the highest rank obtained by an A-student in the spelling bee contest, can be calculated by considering the probabilities of each possible value of H. Since there are 10! possible rankings, the PMF is as follows:
P(H = 1) = 1/10!, P(H = 2) = 9/10!, P(H = 3) = 8/10!, ..., P(H = 10) = 1/10!.
Explanation:
In this problem, we need to determine the probability mass function (PMF) of H, which represents the highest rank obtained by an A-student in the spelling bee contest.
Since there are 10! possible rankings, we need to calculate the probability of each possible value of H.
Let's consider the possible values of H:
- H = 1: This means the A-student obtained the highest rank. There is only one possible ranking where the A-student is ranked first. Therefore, the probability of H = 1 is 1/10!.
- H = 2: This means the A-student obtained the second-highest rank. There are 9 possible rankings where the A-student is ranked second (since the A-student cannot be ranked first). Therefore, the probability of H = 2 is 9/10!.
- H = 3: This means the A-student obtained the third-highest rank. There are 8 possible rankings where the A-student is ranked third (since the A-student cannot be ranked first or second). Therefore, the probability of H = 3 is 8/10!.
- ...
- H = 10: This means the A-student obtained the lowest rank. There is only one possible ranking where the A-student is ranked last. Therefore, the probability of H = 10 is 1/10!.
By calculating the probabilities for each possible value of H, we can determine the probability mass function of H.
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Rewritten by : Jeany
The probability mass function for the highest rank (H) obtained by an A-student is the same for all possible values of H is P(H = h) = 9! / 10! = 1/10
To determine the probability mass function (PMF) of the highest rank (H) obtained by an A-student in the mock spelling bee contest, we need to calculate the probability of each possible value of H.
Since there are 10 students in total (5 from Arapahoe Ridge High and 5 from Boulder High), let's consider the possible values of H:
H = 1: This means the best A-student got the highest rank.
H = 2: This means the best A-student got the second highest rank.
H = 10: This means the best A-student got the lowest rank.
We need to calculate the probability of each of these events happening. Since each possible ranking is equally likely, there are a total of 10! (10 factorial) possible rankings.
Let's calculate the probability for each value of H:
H = 1: The best A-student gets the highest rank, and the other 9 students can be ranked in any order.
Number of favorable outcomes: 9! (since the best A-student is already ranked first)
Number of possible outcomes: 10!
Probability: P(H = 1) = 9! / 10!
H = 2: The best A-student gets the second highest rank, and the other 9 students can be ranked in any order.
Number of favorable outcomes: 9! (since the best A-student is already ranked second)
Number of possible outcomes: 10!
Probability: P(H = 2) = 9! / 10!
H = 10: The best A-student gets the lowest rank (rank 10), and the other 9 students can be ranked in any order.
Number of favorable outcomes: 9! (since the best A-student is already ranked last)
Number of possible outcomes: 10!
Probability: P(H = 10) = 9! / 10!
In general, for H = h, where h is any value from 1 to 10:
P(H = h) = 9! / 10!
This is because the best A-student's rank is fixed at position h, and the remaining 9 students can be ranked in any order.
So, the probability mass function for the highest rank (H) obtained by an A-student is the same for all possible values of H:
P(H = h) = 9! / 10! = 1/10
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