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? C) there will be only one female guest? D) they will have only male guests? E) there will be at least one female guest?
Combinations
A) To have an equal number of males and females at the dinner party, we need to select 4 males and 4 females out of the total of 20 friends.
The number of ways to choose 4 males out of 12 is given by 12C4 = (12!)/(4!(12-4)!) = 495.
The number of ways to choose 4 females out of 8 is given by 8C4 = (8!)/(4!(8-4)!) = 70.
The total number of possible combinations is 20C8 = (20!)/(8!(20-8)!) = 125,970.
Therefore, the probability of having an equal number of males and females at the dinner party is (495*70)/125,970 = 0.275.
B) Since Mandy is one of the 20 friends, the probability of her being among the 8 guests selected can be calculated as 8/20 = 0.4.
C) To have only one female guest, we need to select 1 female out of the 8 available females, and 7 males out of the 12 available males.
The number of ways to choose 1 female out of 8 is given by 8C1 = 8.
The number of ways to choose 7 males out of 12 is given by 12C7 = 792.
Therefore, the probability of having only one female guest is (8*792)/125,970 = 0.0505.
D) To have only male guests, we need to select all 8 guests from the available 12 males.
The number of ways to choose 8 males out of 12 is given by 12C8 = (12!)/(8!(12-8)!) = 495.
Therefore, the probability of having only male guests is 495/125,970 = 0.0039.
E) To calculate the probability of having at least one female guest, we need to find the probability of the complement event (having all male guests) and subtract it from 1.
The probability of having at least one female guest = 1 - probability of having only male guests.
So, the probability of having at least one female guest is 1 - 0.0039 = 0.9961.
To answer these questions, we need to consider the total number of possible combinations when choosing 8 guests from the 20 friends. We can use the formula for combinations:
C(n, r) = n! / (r! * (n-r)!)
Where:
- n is the total number of friends (20 in this case)
- r is the number of guests to be invited (8 in this case)
- ! denotes the factorial of a number, which means multiplying all whole numbers from that number down to 1
Let's proceed with calculating the probabilities now.
A) The probability of having an equal number of male and female guests:
First, we need to calculate the number of combinations where we choose 4 males and 4 females:
C(12, 4) * C(8, 4) = (12! / (4! * 8!)) * (8! / (4! * 4!)) = 12 * 11 * 10 * 9 / (4 * 3 * 2 * 1) * (8 * 7 * 6 * 5 / (4 * 3 * 2 * 1))
Then, we divide this number by the total number of combinations:
C(20, 8) = 20! / (8! * 12!)
Finally, we calculate the probability:
P(A) = C(12, 4) * C(8, 4) / C(20, 8)
B) The probability of inviting Mandy:
Since we know that Mandy is female, we need to calculate the probability of choosing Mandy out of the 8 guests:
C(1, 1) * C(19, 7) = 1 * (19! / (7! * 12!))
Then, we divide this number by the total number of combinations:
C(20, 8) = 20! / (8! * 12!)
Finally, we calculate the probability:
P(B) = C(1, 1) * C(19, 7) / C(20, 8)
C) The probability of having only one female guest:
First, we need to calculate the number of combinations where we choose 1 female and 7 males:
C(8, 1) * C(12, 7) = 8 * (12! / (7! * 5!))
Then, we divide this number by the total number of combinations:
C(20, 8) = 20! / (8! * 12!)
Finally, we calculate the probability:
P(C) = C(8, 1) * C(12, 7) / C(20, 8)
D) The probability of having only male guests:
First, we need to calculate the number of combinations where we choose 8 males:
C(12, 8) = 12! / (8! * 4!)
Then, we divide this number by the total number of combinations:
C(20, 8) = 20! / (8! * 12!)
Finally, we calculate the probability:
P(D) = C(12, 8) / C(20, 8)
E) The probability of having at least one female guest:
To find the probability of the complement event (no female guests), we calculate:
P(E') = C(12, 8) / C(20, 8)
Then, we subtract this probability from 1 to get the probability of at least one female guest:
P(E) = 1 - P(E')
Please note that these calculations assume that all combinations are equally likely.