Question

A box contains four slips of paper marked \(1,2,3\), and 4\. Two slips are selected without replacement. List the possible values for each of the following random variables: a. \(x=\) sum of the two numbers b. \(y=\) difference between the first and second numbers c. \(z=\) number of slips selected that show an even number d. \(w=\) number of slips selected that show a 4

Short answer

a. Possible values of \(x\) are 3,4,5,6,7.\nb. Possible values of \(y\) are -3,-2,-1,0,1,2,3.\nc. Possible values of \(z\) are 0,1,2.\nd. Possible values of \(w\) are 0 and 1.

Step-by-step solution

Sum of the two numbers

For part a, the random variable \(x\) is defined as the sum of the two numbers. The two numbers can be any of the four slips, and since they are selected without replacement, the pair of numbers can be: (1,2), (1,3), (1,4), (2,3), (2,4), (3,4). When summed, the possible values are: 3, 4, 5, 5, 6, and 7.

Difference between the first and second numbers

For part b, the random variable \(y\) is the difference between the first and second numbers, considering the order. If the first number is smaller than the second, the value will be negative, if not, it will be positive. Therefore, the possible values are: -3, -2, -1, 0, 1, 2, 3.

Number of slips selected that show an even number

For part c, the random variable \(z\) is the number of slips selected that show an even number. Since we are selecting 2 slips, we can have 0, 1, or 2 slips showing an even number. Thus, possible values are 0, 1, 2.

Number of slips selected that show a 4

For part d, the random variable \(w\) is the number of slips selected that show a 4. Since we are only selecting 2 slips out of a possible 4, and only one of the 4 slips are labelled as '4', the possible values are 0 (if '4' is not selected) and 1 (if '4' is selected). Therefore, the possible values for \(w\) are 0 and 1.