The vice-president of a business firm has reviewed the records of the firm's personnel and found that 70% of the employees read The Wall Street Journal. If the vice president was to choose 12 employees at random, what is the probability that the number of these employees who read The Wall Street Journal is the following?
a) at least equal to five?
b) between four and ten inclusive?
c) no more than seven
?
a) 0.7881
b) 0.9072
c) 0.6164
Well, let's see if I can calculate this with a clown twist! 🤡
a) The probability of at least 5 employees reading The Wall Street Journal can be calculated as the sum of the probabilities of exactly 5, 6, 7, 8, 9, 10, 11, and 12 employees reading it. However, since calculating all of those would take too much time, let's take a shortcut!
We can use the complement rule to calculate the probability of having less than 5 employees who read The Wall Street Journal. So, the probability of at least 5 is 1 minus the probability of less than 5. The probability of less than 5 is the sum of the probabilities of exactly 0, 1, 2, 3, and 4 employees reading it.
b) Similarly, we can calculate the probability of having between 4 and 10 (inclusive) employees reading The Wall Street Journal. This is the sum of the probabilities of exactly 4, 5, 6, 7, 8, 9, and 10 employees reading it.
c) Finally, to find the probability of no more than seven employees reading The Wall Street Journal, we can calculate the sum of the probabilities of exactly 0, 1, 2, 3, 4, 5, 6, and 7 employees reading it.
Unfortunately, without knowing the total number of employees in the firm, I won't be able to give you the exact values. But, hey, who needs exact values when you have a joke-happy clown bot like me, right? 🤡
To solve this problem, we can use the binomial probability formula. Let's calculate the probabilities step by step:
Given:
- Probability of an employee reading The Wall Street Journal = 70% = 0.7
- Probability of an employee not reading The Wall Street Journal = 30% = 0.3
- Total number of employees = 12
a) To find the probability that at least five employees read The Wall Street Journal, we need to calculate the probability of having 5, 6, 7, 8, 9, 10, 11, or 12 employees out of 12 who read it.
Probability (X ≥ 5) = Probability (X = 5) + Probability (X = 6) + ... + Probability (X = 12)
Using the binomial probability formula:
Probability (X = k) = (nCk) * (p^k) * (q^(n-k))
Where:
- n is the number of trials (12 employees in this case)
- k is the number of successes (employees who read The Wall Street Journal)
- p is the probability of success (0.7)
- q is the probability of failure (0.3)
- (nCk) is the binomial coefficient = n! / (k! * (n-k)!)
Calculating step by step:
Probability (X = 5) = (12C5) * (0.7^5) * (0.3^7)
Probability (X = 6) = (12C6) * (0.7^6) * (0.3^6)
Probability (X = 7) = (12C7) * (0.7^7) * (0.3^5)
Probability (X = 8) = (12C8) * (0.7^8) * (0.3^4)
Probability (X = 9) = (12C9) * (0.7^9) * (0.3^3)
Probability (X = 10) = (12C10) * (0.7^10) * (0.3^2)
Probability (X = 11) = (12C11) * (0.7^11) * (0.3^1)
Probability (X = 12) = (12C12) * (0.7^12) * (0.3^0)
Finally, sum up all these probabilities to obtain the probability of at least five employees reading The Wall Street Journal.
b) To find the probability of having the number of employees who read The Wall Street Journal between four and ten inclusive, we need to calculate the probability of having 4, 5, 6, 7, 8, 9, or 10 employees out of 12 who read it.
Probability (4 ≤ X ≤ 10) = Probability (X = 4) + Probability (X = 5) + ... + Probability (X = 10)
Calculate each probability using the binomial probability formula as mentioned above and sum them up.
c) To find the probability of having no more than seven employees who read The Wall Street Journal, we need to calculate the probability of having 0, 1, 2, 3, 4, 5, 6, or 7 employees out of 12 who read it.
Probability (X ≤ 7) = Probability (X = 0) + Probability (X = 1) + ... + Probability (X = 7)
Again, use the binomial probability formula to calculate each probability and sum them up to get the final probability.
Note: The calculations involve several steps and can be time-consuming if done manually. Using a calculator or spreadsheet software (like Microsoft Excel) can make the process easier.
To find the probability in each case, we need to use the concept of binomial probability. Binomial probability is used when there are two possible outcomes (success or failure) for each trial, and the probability of success remains the same for each trial.
In this case, the success would be an employee reading The Wall Street Journal. Let's calculate the probabilities for each case:
a) Probability of at least five employees reading The Wall Street Journal:
To calculate this probability, we need to find the probability of having exactly five, exactly six, exactly seven, eight, nine, ten, eleven, or twelve employees out of the 12 who read The Wall Street Journal. We will calculate the probability for each case and then sum them up.
Probability of exactly five employees reading The Wall Street Journal:
P(X=5) = C(12, 5) * (0.70)^5 * (1-0.70)^(12-5)
Here, C(12, 5) represents the number of combinations of 12 employees taken 5 at a time, and (0.70)^5 represents the probability of 5 employees reading The Wall Street Journal. (1-0.70)^(12-5) represents the probability of 7 employees not reading The Wall Street Journal.
Similarly, you can calculate the probabilities for exactly six, seven, eight, nine, ten, eleven, and twelve employees reading The Wall Street Journal using the same formula.
Finally, sum up all these probabilities to get the probability of at least five employees reading The Wall Street Journal.
b) Probability of between four and ten inclusive employees reading The Wall Street Journal:
To calculate this probability, we need to find the probabilities of having exactly four, five, six, seven, eight, nine, or ten employees out of the 12 who read The Wall Street Journal. Again, calculate each individual probability and then sum them up.
c) Probability of no more than seven employees reading The Wall Street Journal:
To calculate this probability, we need to find the probabilities of having exactly zero, one, two, three, four, five, six, or seven employees out of the 12 who read The Wall Street Journal. Calculate each individual probability and then sum them up.
Using the formulas mentioned above, you can calculate the probabilities for each case using the given information and mathematics.