Question

a teacher offers gift cards as a reward for classroom participation. the teacher places the gift cards from four different stores into a bag and mixes them well. a student gets to select two gift cards at random (one at a time and without replacement). each outcome in the sample - space for the random selection of two gift cards is equally likely. what is the probability of each outcome in the sample space? $\frac{1}{4}$ $\frac{1}{16}$ $\frac{1}{6}$ $\frac{1}{2}$

Explanation:

Step1: Calculate total number of ways to pick first card

There are 4 choices for the first - card selection.

Step2: Calculate number of ways to pick second card

Since there is no replacement, there are 3 choices for the second - card selection after the first card is picked.

Step3: Calculate total number of outcomes in sample space

By the multiplication principle, the total number of ways to pick 2 cards (one at a time without replacement) is \(4\times3 = 12\) ways.

Step4: Calculate probability of each outcome

Since each outcome is equally likely, and the sum of probabilities of all outcomes in a sample - space is 1, the probability of each outcome is \(\frac{1}{12}\). However, if we consider the problem in terms of combinations (where order of selection doesn't matter), the number of combinations of choosing 2 cards out of 4 (using the combination formula \(C(n,r)=\frac{n!}{r!(n - r)!}\), here \(n = 4\) and \(r=2\), \(C(4,2)=\frac{4!}{2!(4 - 2)!}=\frac{4\times3\times2!}{2!\times2!}=6\)). The probability of each outcome (when order of selection doesn't matter) is \(\frac{1}{6}\).

Answer:

\(\frac{1}{6}\)