grade 12 data management questions

Question

1. A family of 7 are to take a photograph.

a)      How many arrangements are possible if the triplets (Jack, Bob, Don) are side by side    

b)     How many arrangements are possible if the Parents (Aex and Molly) are not side by side

1. George is immersed in a game of Yahtzee. Assuming the game is played with fair dice, find the probability of the following:       

a.      A two or a five with one die.  

b.      Two fives with two dice.                                                                                                               

Answer 1

If there were no restrictions, the number of possible arrangements would be 7! = 5040

a) Consider the triplets to be one entity. Without considering the variations in that group of 3, the number of arrangements is 5! = 120. For each of those, there are 6 ways to order the triplets (3!). Putting that together, we have 120•6 = 720 arrangements

b) Think of arranging all but the father first. There would be 6! = 720 arrangements to that point. Now, putting the father in, there are only 5 positions he can take without being next to the mother. That gets us to 720•5 = 3600 arrangements

a. There are six sides to the die. If it is a fair die, each side has a 1/6 probability of facing up. Since we have two values that count as a success, and they are mutually exclusive, to get the total OR probability, we add 1/6 + 1/6 = 1/3

b. The two dice behave independently. The AND probability of 2 independent events occurring is the product of their individual probabilities. (1/6)(1/6) = 1/36