Consider a class with 10 sophomores, 8 juniors, and 6 seniors. Two students are selected at random.
1) What is the probability of selecting 1 junior and 1 senior.
2) Find the odds of selecting two students who are not seniors.
How do I set these up to solve??
Any help is greatly appreciated!!
To solve these questions, we need to use the concept of probability. Probability is defined as the number of favorable outcomes divided by the total number of possible outcomes.
Let's break down the steps for each question:
1) What is the probability of selecting 1 junior and 1 senior?
Step 1: Determine the total number of possible outcomes.
In this case, the total number of possible outcomes is the total number of ways to select any 2 students out of the entire class, regardless of their grade level. We can calculate this using the combination formula:
Total possible outcomes = nCr = (10 + 8 + 6) C 2 = 24 C 2 = (24 * 23) / (2 * 1) = 276
Step 2: Determine the number of favorable outcomes.
To select 1 junior and 1 senior, we need to choose 1 student out of 8 juniors (favorable outcome 1) and 1 student out of 6 seniors (favorable outcome 2). The number of favorable outcomes is the product of these two:
Favorable outcomes = 8 C 1 * 6 C 1 = 8 * 6 = 48
Step 3: Calculate the probability.
The probability of selecting 1 junior and 1 senior is the number of favorable outcomes divided by the total number of possible outcomes:
P(1 junior and 1 senior) = Favorable outcomes / Total possible outcomes = 48 / 276 ≈ 0.174 (rounded to three decimal places)
Therefore, the probability of selecting 1 junior and 1 senior is approximately 0.174.
2) Find the odds of selecting two students who are not seniors.
The odds are calculated by dividing the number of favorable outcomes by the number of unfavorable outcomes.
Step 1: Determine the number of favorable outcomes.
To select two students who are not seniors, we need to choose both students from the combined pool of sophomores and juniors. The number of favorable outcomes is the number of ways to select 2 students out of the combined pool:
Favorable outcomes = (10 + 8) C 2 = 18 C 2 = (18 * 17) / (2 * 1) = 153
Step 2: Determine the number of unfavorable outcomes.
The number of unfavorable outcomes is the number of ways to select 2 students from the entire class, subtracting the favorable outcomes:
Unfavorable outcomes = Total possible outcomes - Favorable outcomes = 276 - 153 = 123
Step 3: Calculate the odds.
The odds of selecting two students who are not seniors is the number of favorable outcomes divided by the number of unfavorable outcomes:
Odds = Favorable outcomes / Unfavorable outcomes = 153 / 123 = 1.244 (rounded to three decimal places)
Therefore, the odds of selecting two students who are not seniors are approximately 1.244.
To solve these probability problems, we need to use the concept of combinations and calculate the number of favorable outcomes divided by the total number of possible outcomes.
1) Probability of selecting 1 junior and 1 senior:
To find the probability of selecting 1 junior and 1 senior, we need to calculate the number of ways we can choose 1 junior out of 8 juniors and 1 senior out of 6 seniors. We will then divide this by the total number of ways we can choose 2 students out of the entire class, which is (10+8+6)C2.
Number of ways to choose 1 junior out of 8 juniors: 8C1 = 8
Number of ways to choose 1 senior out of 6 seniors: 6C1 = 6
Total number of ways to choose 2 students out of the entire class: (10+8+6)C2 = 24C2 = (24!)/(2!(24-2)!) = (24*23)/(2*1) = 12 * 23 = 276
Therefore, the probability of selecting 1 junior and 1 senior is:
Probability = (Number of favorable outcomes) / (Total number of possible outcomes) = (8C1 * 6C1) / (24C2) = (8 * 6) / 276 = 48 / 276 = 0.1739 or 17.39%
2) Odds of selecting two students who are not seniors:
Odds can be calculated by comparing the number of favorable outcomes to the number of unfavorable outcomes. Here, favorable outcomes refer to the number of ways we can choose 2 students who are not seniors, and unfavorable outcomes refer to the number of ways we can choose 2 students in which at least one of them is a senior.
Number of ways to choose 2 students who are not seniors: (10+8)C2 = 18C2 = (18!)/(2!(18-2)!) = (18*17)/(2*1) = 153
Now, let's calculate the number of unfavorable outcomes. The unfavorable outcomes would be the total number of possible outcomes minus the favorable outcomes. In this case, it would be the number of ways we can choose 2 students out of the entire class (24C2) minus the number of ways we can choose 2 students who are not seniors (18C2).
Number of unfavorable outcomes = (24C2) - (18C2) = 276 - 153 = 123
Therefore, the odds of selecting two students who are not seniors would be:
Odds = (Number of favorable outcomes) : (Number of unfavorable outcomes) = 153 : 123 = 51 : 41
the total students are 24.
Probability of 1junior and 1 senior are..
Pr=(8/24)(6/23)
The second:
Pr=(18/24)(17/23)
Notice in each, the numerator is the number of possible ways to be successful,the denominator is the number of ways it can go.
For number one, there is a total of 24 students when you add 10+8+6. Then you want to find 1 junior and 1 senior which added up becomes 2. soo....
2/24 = 1/12 reduced! :)