Hello! If you don't want tor read a long explanation, skip down below to the next section in bold.
To complete this question, I would use Pascal's triangle to figure out the cases, and then from there complete it with powers. Here's what I mean:
Here's all the rows of Pascal's triangle, up to where we need it if the first one is considered row 0.
0. 1
1. 1 1
2. 1 2 1
3. 1 3 3 1
4. 1 4 6 4 1
5. 1 5 10 10 5 1
6. 1 6 15 20 15 6 1
7. 1 7 21 35 35 21 7 1
As you can see, I stopped listing at row 7, because there are 7 U.S. men. Here's how we can use this row:
1 7 21 35 35 21 7 1
0 1 2 3 4 5 6 7
In this case, the 0, 1, 2, and so on, is the number of men that consider themselves fans, for example, and the numbers above them (from Pascal's triangle) are the number of cases in which that could happen. For example, for the case where 0 men consider themselves fans and 7 don't, then there would be 1 possibility in which that would happen.
So if we look at the number 5 in the second row, as we want the probability 5 men out of 7 consider themselves fans, we'll see there are 21 cases in which 5 men out of 7 consider themselves fans.
Now to apply the probability of 54% to see the chance of one of those 21 cases occurring.
If you were to take the first case, where all 7 men consider themselves not basketball fans, how would you find the probability for that? It would end up being 46% ^ 7, which equals roughly .004, times the number of total cases (numbers in the first row), in this case 1.
You can do the same for the second case, where 1 man considers themselves a basketball fan and 2 don't- 46% ^ 6 * 54% ^ 1 * 7 [number above 1 in the two rows above] equaling roughly .036.
Now, to apply it to the case we actually are looking at. Skip here if you don't want to read explanations and just want the actual calculation.
5 men consider themselves fans, 2 don't.
54% ^ 5 * 46% ^ 2 * 21 = 0.0459165024 * 0.2116 * 21 = 0.20403457 = roughly 20.4%
So your answer is, the probability that 5 men randomly selected from a group of 7 U.S. men consider themselves basketball fans is 20.4%.
Answer "and" Explanation:
To find the probability, we first need to find the correct binomial coefficient from Pascal’s Triangle. We’re working with a sample size of 7 and picking 5 successes (which in this case is a man that is a basketball fan), so we need to find the element at the 5th index of the 7th row.
When we generate the triangle, we find that the 7th row looks like this:
{1, 7, 21, 35, 35, 21, 7, 1}
We number the elements from 0, so the 5th index is the 6th element, or 21.
Now, we can use the probabilities. Our probability of success is 0.54. This means that our probability of failure is 0.46. Because we want 5 successes, we will then have 2 failures. When we set up the expression, we get:
[tex]21(0.54)^{5} (0.46)^{2}[/tex]
Now, put this in a calculator to get a probability of 20.4%
So the answer will be: 20.4%
