A) To have an equal number of males and females at the dinner party, they need to choose 4 males and 4 females out of their 20 good friends. The total number of ways they can choose 4 males out of 12 is given by the combination formula: C(12, 4) = 12! / (4! * (12-4)!) = 495.
Similarly, the total number of ways they can choose 4 females out of 8 is given by the combination formula: C(8, 4) = 8! / (4! * (8-4)!) = 70.
Therefore, the total number of ways they can have an equal number of males and females is: 495 * 70 = 34,650.
The total number of ways they can invite any 8 friends out of their 20 is given by the combination formula: C(20, 8) = 20! / (8! * (20-8)!) = 125,970.
So, the probability that there will be an equal number of males and females at the dinner party is: 34,650 / 125,970 = 0.2757 (rounded to four decimal places).
B) The probability of Mandy being among the invited guests is 1 out of 20, as there is only one Mandy among their 20 good friends. Therefore, the probability that Mandy will be among these invited is 1 / 20 = 0.05 (rounded to two decimal places).
C) To have only one female guest, they need to choose 7 males and 1 female out of their 20 good friends. The total number of ways they can choose 7 males out of 12 is given by the combination formula: C(12, 7) = 12! / (7! * (12-7)!) = 792.
Similarly, the total number of ways they can choose 1 female out of 8 is given by the combination formula: C(8, 1) = 8! / (1! * (8-1)!) = 8.
Therefore, the total number of ways they can have only one female guest is: 792 * 8 = 6,336.
So, the probability that there will be only one female guest is: 6,336 / 125,970 = 0.0503 (rounded to four decimal places).
D) To have only male guests, they need to choose 8 males out of their 12 male friends. The total number of ways they can choose 8 males out of 12 is given by the combination formula: C(12, 8) = 12! / (8! * (12-8)!) = 495.
Therefore, the total number of ways they can have only male guests is: 495.
So, the probability that they will have only male guests is: 495 / 125,970 = 0.0039 (rounded to four decimal places).
E) The probability of having at least one female guest can be calculated by subtracting the probability of having only male guests from 1. So, the probability of having at least one female guest is: 1 - 0.0039 = 0.9961 (rounded to four decimal places).
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?
3 answers
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