A hat contains 20 names, 9 of which are females. If 8 names are randomly drawn from the hat, what is the probability that at least 6 male names are drawn??????
So we have 9 females and 11 males,
we are choosing 8
the number of cases you are interested in are:
at least 6 males ---->
6 males, 2 females + 7males, 1 female + 8 males
= C(11,6)C(9,2) + C(11,7)C(9,1) + C(11,8)
= 462(36) + 330(9) + 165
= 19767
number of way to choose 8 of 20 = C(20,8) = 125970
so prob (at least 6 men) = 19767/125970
To find the probability of at least 6 male names being drawn, we need to consider the different possible outcomes and calculate the probability of each.
First, let's calculate the total number of possible outcomes. We are drawing 8 names from a hat containing 20 names, so the total number of possible outcomes is given by the combination formula:
C(n, r) = n! / (r!(n-r)!)
Where n is the total number of items and r is the number of items being chosen. In this case, n = 20 and r = 8. Plugging these values into the formula, we get:
C(20, 8) = 20! / (8!(20-8)!)
Next, let's calculate the number of outcomes where at least 6 male names are drawn.
Case 1: Exactly 6 male names are drawn
We choose 6 male names out of the 9 available and 2 female names out of the 11 remaining. This can be calculated using the combination formula:
C(9, 6) * C(11, 2)
Case 2: Exactly 7 male names are drawn
We choose 7 male names out of the 9 available and 1 female name out of the 11 remaining. Again, using the combination formula:
C(9, 7) * C(11, 1)
Case 3: All 8 names drawn are male
We choose all 8 male names out of the 9 available. This can be calculated using the combination formula:
C(9, 8)
Now, we can calculate the probability of each case by dividing the number of outcomes for each case by the total number of possible outcomes:
Probability = (Number of desired outcomes) / (Total number of possible outcomes)
Finally, add up the probabilities of the three cases to get the probability of at least 6 male names being drawn.
I hope this explanation helps!