1. A pair of fair dice is thrown. If the two numbers appearing are different, find the probability p that:
(a) the sum is 6; (b) an ace appears; (c) the sum is 4 or less.
There are 36 ways the pair of dice can be thrown, and six of them, (1,1),(2,2),…,(6,6), have the same numbers. Thus the reduced sample space will consist of 36−6=30 elements.
(a) The sum 6 can appear in four ways: (1,5),(2,4),(4,2),(5,1). (We cannot include (3,3) since the numbers are the same.) Hence p=304=152.
(b) An ace can appear in 10 ways: (1,2),(1,3),…,(1,6) and (2,1),(3,1),…,(6,1). Therefore p=3010=31.
(c) The sum of 4 or less can occur in four ways: (3,1),(1,3),(2,1),(1,2). Thus p=304=152.
7.22. A class has 12 boys and four girls. Suppose three students are selected at random from the class. Find the probability p that they are all boys.
The probability that the first student selected is a boy is 12/16 since there are 12 boys out of 16 students. If the first student is a boy, then the probability that the second is a boy is 11/15 since there are 11 boys left out of 15 students. Finally, if the first two students selected were boys, then the probability that the third student is a boy is 10/14 since there are 10 boys left out of 14 students. Thus, by the multiplication theorem, the probability that all three are boys is p=1612⋅1511⋅1410=2811 $
(Probability, Dice, Sample Space, Ace, Contingency)
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Step-by-Step Solution
Step 1: Determine total number of students
First, we need to find the total number of students in the class. This is the sum of the number of boys and the number of girls.
Step 2: Probability of selecting the first boy
The probability of selecting the first boy is the ratio of the number of boys to the total number of students. After selecting one boy, both the number of boys and the total number of students decrease by one.
Step 3: Probability of selecting the second boy
After one boy has been selected, there are now 11 boys left and a total of 15 students remaining. The probability of selecting a second boy is based on these new counts.
Step 4: Probability of selecting the third boy
Following the selection of two boys, there are 10 boys left and 14 students in total. This gives us the probability of selecting a third boy.
Step 5: Calculate the total probability
To find the probability that all three selected students are boys, we multiply the probabilities of each sequential selection. This is an application of the multiplication theorem for dependent events.