1) There are 4 cardio stations and 6 weight lifting stations at the gym. How many ways can you choose 2 cardio and 2 weight lifting stations for your workout?
2)A diner allows you to choose three fillings for their omelets from a list of 10. How many different omelets can be made using three different fillings? (like 10P2 or 25C25)
3) There are 4 cardio stations and 6 weight lifting stations at the gym. You decide to choose 4 stations randomly for today's workout. What is the probability you choose all weight lifting or all cardio stations?
4)There were 5 women and 6 men who were interviewed for five positions. Their qualifications experience, and interviews were very similar. One woman and four men were hired. Is it reasonable that this could have simply happened by chance or is this evidence that the women may have been discriminated against? Support your conclusion with a statistical argument.
5) A cake recipe calls for combining four dry ingredients (flour, sugar, salt, baking powder). In how many orders can you mix any three of the four dry ingredients? (like 10P2 or 25C25)
Thanks!
1) To calculate the number of ways you can choose 2 cardio and 2 weight lifting stations for your workout, you can use the combination formula.
The number of ways to choose 2 cardio stations from the 4 available cardio stations is given by 4C2, which is equal to (4!)/(2!(4-2)!) = 6.
Similarly, the number of ways to choose 2 weight lifting stations from the 6 available weight lifting stations is given by 6C2, which is equal to (6!)/(2!(6-2)!) = 15.
To find the total number of ways to choose 2 cardio and 2 weight lifting stations, you need to multiply the number of ways to choose cardio stations by the number of ways to choose weight lifting stations.
Therefore, the total number of ways is 6 * 15 = 90.
2) To calculate the number of different omelets that can be made using three different fillings, you can use the combination formula.
The number of ways to choose 3 fillings from a list of 10 is given by 10C3, which is equal to (10!)/(3!(10-3)!) = 120.
Therefore, there are 120 different omelets that can be made using three different fillings.
3) To calculate the probability of choosing all weight lifting or all cardio stations when randomly choosing 4 stations, you need to find the number of successful outcomes (choosing all weight lifting or all cardio stations) divided by the total number of possible outcomes (choosing any 4 stations).
The number of successful outcomes of choosing all weight lifting stations is equal to 1, as there is only one way to choose all 4 weight lifting stations.
Similarly, the number of successful outcomes of choosing all cardio stations is also equal to 1, as there is only one way to choose all 4 cardio stations.
The total number of possible outcomes of choosing any 4 stations is calculated by finding the number of ways to choose 4 stations from the total number of stations at the gym, which is 10. This is equal to 10C4 = (10!)/(4!(10-4)!) = 210.
Therefore, the probability of choosing all weight lifting or all cardio stations is (1 + 1)/210 = 2/210 = 1/105.
4) To determine whether the hiring process is showing evidence of discrimination or happening purely by chance, you can use statistical analysis. One way to analyze this is by conducting a hypothesis test.
We can set up a hypothesis test where the null hypothesis (H0) states that there is no discrimination and the alternative hypothesis (Ha) states that there is discrimination against women.
If the 5 positions were randomly assigned, the probability of exactly one woman being hired and four men being hired can be calculated using binomial probability. The probability of hiring one specific woman is (1/11) and the probability of hiring four specific men is (6/11)^4. Since there are 5 women to choose from and 6 men to choose from, we can multiply these probabilities by the number of combinations, which is 5C1 * 6C4.
The probability of this outcome happening purely by chance can be calculated as follows:
P(chance) = (5C1 * 6C4) * (1/11) * (6/11)^4 ≈ 0.351
If this probability is low (below a predetermined significance level, such as 0.05), we can reject the null hypothesis in favor of the alternative hypothesis. This would suggest evidence of discrimination against women in the hiring process. If the probability is high, we fail to reject the null hypothesis and conclude that the hiring outcome could have happened by chance.
5) To calculate the number of orders you can mix any three of the four dry ingredients, you can use the permutation formula.
The number of ways to choose 3 out of the 4 ingredients can be found using 4P3, which is equal to 4!/(4-3)! = 4!/1! = 4.
Therefore, you can mix the three dry ingredients in 4 different orders.
1) To calculate the number of ways you can choose 2 cardio and 2 weight lifting stations for your workout, you can use the combination formula. The formula for calculating combinations is:
nCr = n! / (r!(n-r)!)
Here, n represents the total number of stations available, and r represents the number of stations you want to choose.
For the given scenario:
Number of cardio stations (n1) = 4
Number of weight lifting stations (n2) = 6
Number of cardio stations to choose (r1) = 2
Number of weight lifting stations to choose (r2) = 2
The number of ways to choose 2 cardio and 2 weight lifting stations can be calculated as follows:
nCr = n1Cr1 * n2Cr2
= 4C2 * 6C2
Now, let's calculate each combination step by step:
4C2 = 4! / (2!(4-2)!)
= (4 * 3) / (2 * 1)
= 6
6C2 = 6! / (2!(6-2)!)
= (6 * 5) / (2 * 1)
= 15
Finally, multiplying these two combinations together:
4C2 * 6C2 = 6 * 15
= 90
So, there are 90 ways to choose 2 cardio and 2 weight lifting stations for your workout.
2) To calculate the number of different omelets that can be made using three different fillings out of a list of 10, you can use the combination formula. Again, the formula for combinations is:
nCr = n! / (r!(n-r)!)
In this case:
Number of fillings available (n) = 10
Number of fillings to choose (r) = 3
The number of different omelets can be calculated as:
nCr = 10C3
Using the combination formula:
10C3 = 10! / (3!(10-3)!)
= (10 * 9 * 8) / (3 * 2 * 1)
= 120
So, there are 120 different omelets that can be made using three different fillings from a list of 10.
3) To calculate the probability of choosing all weight lifting or all cardio stations out of the total available stations, you need to determine the number of favorable outcomes (choosing all weightlifting or all cardio stations) and divide it by the total number of possible outcomes (choosing any 4 stations).
Number of cardio stations (n1) = 4
Number of weight lifting stations (n2) = 6
Total stations to choose (r) = 4
To find the number of favorable outcomes:
For all cardio stations:
Number of ways to choose all cardio = 1 (as you can only choose all cardio stations once)
For all weight lifting stations:
Number of ways to choose all weight lifting = 1 (as you can only choose all weight lifting stations once)
So, the number of favorable outcomes = 1 + 1 = 2
To find the total number of possible outcomes, we calculate the combination of all stations:
Total number of possible outcomes = (n1+n2)Cr = (4+6)C4 = 10C4
Using the combination formula:
10C4 = 10! / (4!(10-4)!)
= 210
Finally, calculate the probability:
Probability = Number of favorable outcomes / Total number of possible outcomes
= 2 / 210
= 1/105
≈ 0.0095
So, the probability of choosing all weight lifting or all cardio stations is approximately 0.0095.
4) To determine if the outcome of hiring one woman and four men out of the total pool of candidates is reasonable or evidence of discrimination, we can use statistical analysis. We will compare the observed outcome with the expected outcome if there was no discrimination.
First, let's calculate the total number of ways to hire 5 people out of the pool of 5 women and 6 men. This can be calculated as a combination:
Total number of ways = Total candidatesCpositions to fill
= (5+6)C5
= 11C5
Using the combination formula:
11C5 = 11! / (5!(11-5)!)
= 462
This represents the total number of possible outcomes if hiring was conducted purely based on qualifications and without discrimination.
Now, let's calculate the number of favorable outcomes for hiring one woman and four men. Since the number of favorable outcomes is 1, it means that only one specific combination fits this criteria.
The probability of hiring one woman and four men due to chance is given by:
Probability = Number of favorable outcomes / Total number of possible outcomes
= 1 / 462
≈ 0.0022
If the observed outcome is close to or greater than this probability, it suggests that the outcome could have simply happened by chance. However, if the observed outcome is significantly lower than this probability, it may indicate possible discrimination.
Therefore, you would need to compare the observed outcome with this probability to determine if the hiring outcome is reasonable or if there is evidence of discrimination.
5) To calculate the number of orders in which you can mix any three of the four dry ingredients (flour, sugar, salt, baking powder), you can use the permutation formula. The formula for calculating permutations is:
nP r = n! / (n-r)!
Here, n represents the total number of ingredients, and r represents the number of ingredients you want to choose.
For the given scenario:
Number of dry ingredients (n) = 4
Number of dry ingredients to choose (r) = 3
The number of orders to mix any three of the four dry ingredients can be calculated as follows:
nP r = 4P3
Using the permutation formula:
4P3 = 4! / (4-3)!
= 4! / 1!
= 4 * 3 * 2
= 24
So, there are 24 different orders in which you can mix any three of the four dry ingredients.