Suppose that two employees have to be fired from a group of seven:
• Amy, female, age 35
• Bob, male, age 41
• Chad, male, age 43
• Diana, female, age 44
• Elvis, male, age 49
• Frank, male, age 61
• Ginger, female, age 62
Determine the probability (both as a fraction and a decimal) that the average age of the employees selected is older than 50.
d. Determine the probability that both employees selected are older than 6
xBar > 50
=> {49+61, 49+62, 61+62}
=>
P(xBar>50)=3/7C2=3/28
xBar>60
=> {61+62}
=>
P(xBar>60)=1/7C2=1/28
To determine the probability that the average age of the selected employees is older than 50, we need to consider the possible combinations of two employees that can be selected.
First, let's find the total number of possible combinations of two employees that can be selected from the group of seven. This can be calculated using the formula for combinations:
C(n, r) = n! / (r!(n-r)!),
where n is the total number of items to choose from and r is the number of items to be chosen.
In this case, n = 7 (the total number of employees) and r = 2 (the number of employees to be selected). So,
C(7, 2) = 7! / (2!(7-2)!) = 7! / (2!5!) = (7 × 6 × 5!) / (2 × 1 × 5!) = 7 × 6 / 2 × 1 = 21.
Therefore, there are 21 possible combinations of two employees that can be selected.
Next, let's determine the number of combinations where the average age of the selected employees is older than 50.
From the given ages, we can identify the employees who are older than 50, which are Frank (age 61) and Ginger (age 62). There are 2 employees who are older than 50.
Now, we need to calculate the number of combinations of 2 employees that include both Frank and Ginger. Again, we use the combination formula:
C(2, 2) = 2! / (2!(2-2)!) = 2! / (2!0!) = (2 × 1) / (2 × 1) = 1.
Therefore, there is only 1 combination where both Frank and Ginger are selected.
Finally, we can calculate the probability that the average age of the selected employees is older than 50 by dividing the number of combinations where the average age is older than 50 by the total number of possible combinations:
Probability = Number of combinations where average age > 50 / Total number of combinations = 1 / 21.
As a fraction, the probability is 1/21, and as a decimal, the probability is approximately 0.048 (rounded to three decimal places).
Now, let's determine the probability that both employees selected are older than 60.
From the given ages, we can identify the employees who are older than 60, which are Frank (age 61) and Ginger (age 62). There are 2 employees who are older than 60.
To find the number of combinations where both Frank and Ginger are selected, we use the same combination formula:
C(2, 2) = 2! / (2!(2-2)!) = 1.
Therefore, there is only 1 combination where both Frank and Ginger are selected.
The probability that both employees selected are older than 60 is given by:
Probability = Number of combinations where both employees > 60 / Total number of combinations = 1 / 21.
As a fraction, the probability is 1/21, and as a decimal, the probability is approximately 0.048 (rounded to three decimal places).
To determine the probability that the average age of the employees selected is older than 50, we need to calculate the total number of possible combinations and the number of combinations that meet the criteria.
Firstly, let's identify the number of possible combinations when two employees are selected from a group of seven. This can be calculated using the formula for combinations, also known as "n choose r":
nCr = n! / [(n - r)! * r!]
In this case, n (the total number of employees) is 7, and r (the number of employees to be selected) is 2. So we have:
7C2 = 7! / [(7 - 2)! * 2!] = 7! / [5! * 2!] = (7 * 6 * 5!) / (5! * 2 * 1) = 21
So there are 21 possible combinations when two employees are selected from the group of seven.
Now, let's determine the number of combinations where the average age is older than 50. We can identify these combinations by listing all the pairs of employees with an average age greater than 50:
Elvis (49) and Frank (61)
Elvis (49) and Ginger (62)
Frank (61) and Ginger (62)
So there are 3 combinations where the average age is older than 50.
Therefore, the probability (as a fraction) that the average age of the employees selected is older than 50 is:
Probability = Number of Desired Outcomes / Total Number of Possible Outcomes
Probability = 3/21
To express this probability as a decimal, divide the numerator (3) by the denominator (21):
Probability = 3/21 = 0.14285714285714285
So, the probability (as a decimal) that the average age of the employees selected is older than 50 is approximately 0.1429.
Moving on to the second question, we need to determine the probability that both employees selected are older than 60.
From the given information, we can see that there are no employees older than 60 in the group. Therefore, the probability of selecting two employees older than 60 is zero.