Among the contestants in a competition are 44 women and 26 men. if 2 winners are randomly selected, what is the probabilty that they are all men?
that's simple ...
prob = (26/70)(25/69) = 65/483
or in combination notation:
C(26,2)/C(70,2) = 325/2415 = 65/483
To find the probability that both winners are men, we need to determine the total number of possible outcomes and the number of favorable outcomes.
Total Number of Possible Outcomes:
Since 2 winners are randomly selected, the total number of possible outcomes is given by the combination formula:
Total number of possible outcomes = nCr(total number of contestants, number of winners selected)
In this case, there are 70 contestants in total (44 women + 26 men) and 2 winners will be selected:
Total number of possible outcomes = 70C2
Favorable Outcomes:
We want to find the number of favorable outcomes where both winners are men. Since there are 26 men in the competition, we need to choose 2 men from the group of 26:
Number of favorable outcomes = 26C2
Probability Calculation:
The probability of an event occurring is given by the ratio of the number of favorable outcomes to the total number of possible outcomes.
Probability of selecting 2 men = Number of favorable outcomes / Total number of possible outcomes
Probability = 26C2 / 70C2
You can simplify the calculation by using the combination formula:
nCr(n, r) = n! / (r! * (n-r)!)
So the simplified probability calculation is:
Probability = (26! / (2! * (26-2)!)) / (70! / (2! * (70-2)!))
After evaluating this expression, you will get the probability that both winners will be men.