Question

question 12
there are 3 cards in a hat: one is a king, one is a queen, and one is an ace. two cards are to be selected at random with replacement. using a tree diagram, obtain the sample space for the experiment. list the elements that make up the sample space.
a kq, ka, qk, qa, ak, aq
b kk, kq, ka, qq, qa, aq, aa
c kk, kq, ka, qk, qq, qa, ak, aq, aa
d kko, kak, qqq, kq, aak

Explanation:

Step1: Analyze the sample space elements

The sample space should list all possible combinations of two - card selections from the three - card set (king \(K\), queen \(Q\), ace \(A\)). When selecting two cards with replacement, the possible ordered pairs (first card, second card) are considered.
The formula for the number of elements in the sample space when selecting \(n\) elements from \(m\) elements with replacement is \(m\times m\). Here \(m = 3\), so there are \(3\times3=9\) elements.
The elements of the sample space are \((K,Q),(K,A),(Q,K),(Q,A),(A,K),(A,Q),(K,K),(Q,Q),(A,A)\). In the given options, option B lists the correct set of ordered pairs.

Answer:

B. KK, KQ, KA, QQ, QA, AA, AQ, AK, KQ