Problem 1 A family consisting of 2 parents, 4 children, and 1 dog is taking a pi

Question

Problem 1 A family consisting of 2 parents, 4 children, and 1 dog is taking a picture with all 7 family members standing in a row. (a) Find the probability that the two parents are not next to each other. (b) Find the probability that the dog is standing somewhere to the right of all 4 children.

Problem 2 In a group of 200 students, 73 speak Spanish, 63 speak Italian, and 56 speak French. Additionally, 22 speak Spanish and Italian, 18 speak Spanish and French, 15 speak Italian and French, and 5 speak all three. Find the probability that a randomly chosen student speaks exactly one of the three languages.

Problem 3 A standard 52-card deck contains 13 ranks (A, 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K) and 4 suits ( ): A , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , J , Q , K , A , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , J , Q , K , A , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , J , Q , K , A , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , J , Q , K .

(a) Find the probability that a 5 card hand is missing at least one suit. example: 2 , 4 , J , 10 , K (missing ) example: 2 , 4 , J , 10 , K (missing ) (b) Find the probability that a 10-card hand contains one 3-of-a-kind, three pairs, and one single (for a total of f ive different ranks). example: 6 , 6 , 6 , 2 , 2 , 9, 9, J, J, 5.

Explanation / Answer

SOlving question 1.

a) P( Two parents are not next to each other= 1- P( next to each other) = 1- (6!*2/7!)=5/7

b) P( dog to the right of 4 chilrden )= Ways in which dog is on extreme right + 2nd step+ 3rd step = [6! + 5!*2 + 4!* 2 ]/7! =0.2

Please post the second individually

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