A poll was taken of 10,439 working adults aged 40-70 to determine their level of education. The participants were classified by sex and by level of education. The results are shown below.
Education Level Male Female Total
High School or Less 2023 3264 5287
Bachelor's Degree 2225 2113 4338
Master's Degree 416 305 721
Ph.D. 45 48 93
Total 4709 5730 10,439
What is the probability that the selected person has a Ph.D., given that she is female?
What is the probability that the selected person is male, given that he has a Master's degree?
48/5730 = ?
416/721 = ?
What virus?
To find the probability that the selected person has a Ph.D., given that she is female, we will use the formula for conditional probability:
P(Ph.D. | Female) = P(Ph.D. and Female) / P(Female)
We can see from the table that there are 93 people with a Ph.D. in total, and 48 of them are female:
P(Ph.D. and Female) = 48 / 10,439
We also know that there are a total of 5,730 female participants:
P(Female) = 5,730 / 10,439
Now we can calculate the probability:
P(Ph.D. | Female) = (48 / 10,439) / (5,730 / 10,439)
Simplifying the expression:
P(Ph.D. | Female) = (48 / 10,439) * (10,439 / 5,730)
P(Ph.D. | Female) = 48 / 5,730
P(Ph.D. | Female) ≈ 0.00840
Therefore, the probability that the selected person has a Ph.D., given that she is female, is approximately 0.00840.
Now, let's move on to finding the probability that the selected person is male, given that he has a Master's degree.
Using the same conditional probability formula:
P(Male | Master's degree) = P(Male and Master's degree) / P(Master's degree)
From the table, we can see that there are 721 participants with a Master's degree in total, and 416 of them are male:
P(Male and Master's degree) = 416 / 10,439
We also know that there are a total of 4,709 male participants:
P(Male) = 4,709 / 10,439
Now we can calculate the probability:
P(Male | Master's degree) = (416 / 10,439) / (4,709 / 10,439)
Simplifying the expression:
P(Male | Master's degree) = (416 / 10,439) * (10,439 / 4,709)
P(Male | Master's degree) = 416 / 4,709
P(Male | Master's degree) ≈ 0.08823
Therefore, the probability that the selected person is male, given that he has a Master's degree, is approximately 0.08823.
To answer these questions, we can use conditional probability, which relates the probability of two events happening together.
For the first question, we are looking for the probability that a selected person has a Ph.D., given that she is female. We can use the equation for conditional probability:
P(Ph.D. | Female) = P(Ph.D. and Female) / P(Female)
To find P(Ph.D. and Female), we use the numbers provided in the table: 48 females have a Ph.D. So, P(Ph.D. and Female) = 48/10,439.
To find P(Female), we can sum up the number of females in the table: 5730. So, P(Female) = 5730/10,439.
Plugging these values into the conditional probability formula:
P(Ph.D. | Female) = 48/10,439 / 5730/10,439 = 48/5730 ≈ 0.008375
Therefore, the probability that the selected person has a Ph.D., given that she is female is approximately 0.008375.
For the second question, we are looking for the probability that a selected person is male, given that he has a Master's degree. Again, we will use conditional probability:
P(Male | Master's Degree) = P(Male and Master's Degree) / P(Master's Degree)
Using the numbers from the table, we can see that 416 people have a Master's degree, and out of those, 416 are males. So, P(Male and Master's Degree) = 416/10,439.
And P(Master's Degree) is just the number of people with a Master's degree, which is 416/10,439.
Plugging these values into the conditional probability formula:
P(Male | Master's Degree) = 416/10,439 / 416/10,439 = 1
Therefore, the probability that the selected person is male, given that he has a Master's degree is 1 or 100%.