Rain is an amazing singer, but cannot play any instruments. He desperately wants a rock band that has a drummer, a guitarist, a bass player, and himself on vocals. He decides to post an ad at the local coffee house for open auditions for his band - The Combinations.
At the auditions, Rain gets 5 females and 9 males trying out for the remaining 3 positions in the band that can play all the instruments.
a) What is the probability that the band will have 3 males and 1 female?
b) What is the probability that the band will have at least 2 males?
Follow the logic i gave you in your previous post. Remember, the formula for n-choose-x is n!/x!*(n-x)! where ! means factorial.
hint for b) Since Rain is male, having at least two males means picking at least one male. Which is equal to (1-P) where P is the probability of picking 3 females.
To solve these probability problems, we need to consider the total number of possible combinations and the number of favorable outcomes.
Let's start with part (a):
a) The total number of possible combinations of 3 males and 1 female from the available auditions can be calculated using the combination formula:
C(n, k) = n! / (k! * (n-k)!)
where n is the total number of available auditions (14 in this case) and k is the number of auditions we want to consider (3 males).
Using this formula, we can calculate the total number of combinations:
C(14, 3) = 14! / (3! * (14-3)!) = 14! / (3! * 11!) = (14 * 13 * 12) / (3 * 2 * 1) = 364
Now, let's consider the number of favorable outcomes, which is the number of ways to choose 3 males out of the 9 available males, multiplied by the number of ways to choose 1 female out of the 5 available females:
Number of favorable outcomes = C(9, 3) * C(5, 1) = (9! / (3! * (9-3)!) * 5! / (1! * (5-1)!)) = (9! / (3! * 6!)) * (5! / (1! * 4!)) = (9 * 8 * 7) / (3 * 2 * 1) * (5 * 4) / (1 * 1) = 84 * 20 = 1680
Therefore, the probability that the band will have 3 males and 1 female is:
P(3 males and 1 female) = (Number of favorable outcomes) / (Total number of combinations) = 1680 / 364 = 4.62 (rounded to two decimal places)
The answer is approximately 4.62%.
Now let's move on to part (b):
b) To calculate the probability that the band will have at least 2 males, we need to find the number of favorable outcomes where the band has 2 or 3 males.
Number of favorable outcomes with 2 males = C(9, 2) * C(5, 2) = (9! / (2! * (9-2)!) * 5! / (2! * (5-2)!)) = (9 * 8 / (2 * 1)) * (5 * 4 / (2 * 1)) = 36 * 10 = 360
Number of favorable outcomes with 3 males = C(9, 3) * C(5, 0) = (9! / (3! * (9-3)!) * 5! / (0! * (5-0)!)) = (9! / (3! * 6!)) * (5! / (1! * 5!)) = (9 * 8 * 7 / (3 * 2 * 1)) * 1 = 84
Total number of favorable outcomes = Number of favorable outcomes with 2 males + Number of favorable outcomes with 3 males = 360 + 84 = 444
Therefore, the probability that the band will have at least 2 males is:
P(at least 2 males) = (Number of favorable outcomes) / (Total number of combinations) = 444 / 364 = 1.22 (rounded to two decimal places)
The answer is approximately 1.22%.