Matt and his wife have 20 good friends 12 of them are male and eight of them or female. They decide to have a dinner party but can invite only ate guess they decide to invite their guess by giant in their names from a hat. What is the probability that A) there will be an equal number of males and females at the dinner party? B) Mandy will be among these invited? Combinations
To solve this problem, we need to use combinations.
A) Probability of equal number of males and females:
There are 12 males and 8 females among their friends. They want to invite 8 guests, so they need to choose 4 males and 4 females. The number of ways to choose 4 males from 12 is given by the combination formula: C(12,4).
Similarly, the number of ways to choose 4 females from 8 is given by the combination formula: C(8,4).
To calculate the probability, we need to divide the number of favorable outcomes (choosing an equal number of males and females) by the total number of possible combinations.
Total number of possible combinations = C(20,8) (choosing 8 guests from 20 friends).
Therefore, the probability of having an equal number of males and females at the dinner party is:
P(equal number) = (C(12,4) * C(8,4)) / C(20,8)
B) Probability of Mandy being invited:
Mandy's probability of being invited can be calculated by dividing the number of favorable outcomes (where Mandy is invited) by the total number of possible combinations.
The number of ways Mandy can be invited is 1 (since her name is fixed), and the remaining 7 guests can be chosen from the other 19 friends (excluding Mandy). Therefore, the total number of possible combinations is C(19,7).
So, the probability of Mandy being invited is:
P(Mandy) = 1 / C(19,7)
A) To calculate the probability of having an equal number of males and females at the dinner party, we need to find the number of favorable outcomes divided by the total number of possible outcomes.
Favorable outcomes:
We can choose 4 males out of the 12 available and 4 females out of the 8 available. This can be done using combinations.
Number of ways to choose 4 males from 12: C(12, 4) = 12! / (4! * (12-4)!) = 495
Number of ways to choose 4 females from 8: C(8, 4) = 8! / (4! * (8-4)!) = 70
Total number of possible outcomes:
We need to choose 8 guests from a total of 20. This can be done using combinations.
Total number of ways to choose 8 guests from 20: C(20, 8) = 20! / (8! * (20-8)!) = 125,970
Therefore, the probability that there will be an equal number of males and females at the dinner party is: (495 * 70) / 125,970 ≈ 0.275 or 27.5%
B) To calculate the probability of Mandy being invited, we need to find the number of favorable outcomes divided by the total number of possible outcomes.
Favorable outcomes:
Mandy needs to be one of the 8 guests chosen. Therefore, the number of favorable outcomes is 1.
Total number of possible outcomes:
We need to choose 8 guests from a total of 20. This can be done using combinations.
Total number of ways to choose 8 guests from 20: C(20, 8) = 20! / (8! * (20-8)!) = 125,970
Therefore, the probability that Mandy will be among the invited guests is: 1 / 125,970 ≈ 0.0000079 or 0.00079%