Question

slips of paper marked with the numbers 1, 2, 3, and 4 are placed in a box. after being mixed, two slips are drawn simultaneously. write out the sample space s, choosing an s with equally likely outcomes, if possible. then give the value of n(s) and tell whether the outcomes in s are equally likely. finally, write the indicated events below in set notation.
a. both slips are marked with even numbers.
b. one slip is marked with an odd number and the other is marked with an even number.
c. both slips are marked with the same number.

what is the sample space?
a. s = {(1,3)}
b. s = {(1,2), (1,3), (1,4), (2,3), (2,4), (3,4)}
c. s = {(1,2), (1,4), (3,2), (3,4)}
d. s = {1, 2, 3, 4}

Explanation:

Step1: Understand Sample Space for Drawing Two Slips

When drawing two slips simultaneously from numbers 1, 2, 3, 4, we consider all possible unordered pairs (since drawing (1,2) is the same as (2,1) when drawn simultaneously). The combinations of two distinct numbers from 1,2,3,4 are calculated using combinations (since order doesn't matter in simultaneous draws). The formula for combinations is \( C(n,k)=\frac{n!}{k!(n - k)!} \), where \( n = 4 \) (total numbers) and \( k = 2 \) (slips drawn). So \( C(4,2)=\frac{4!}{2!(4 - 2)!}=\frac{4\times3\times2!}{2!\times2!}=\frac{12}{2}=6 \). The possible pairs are (1,2), (1,3), (1,4), (2,3), (2,4), (3,4).

Step2: Analyze Each Option

• Option A: Only (1,3) is given, which is not all possible pairs. Incorrect.
• Option B: Contains all 6 possible unordered pairs of two distinct numbers from 1,2,3,4. Correct.
• Option C: Only contains pairs with odd and even, missing pairs like (1,3), (2,3). Incorrect.
• Option D: Is the set of individual numbers, not pairs. Incorrect.

Answer:

B. \( S = \{(1,2), (1,3), (1,4), (2,3), (2,4), (3,4)\} \)