Question

1. in a closet, there are 4 different colored tops: red, green, purple, and black. there are also shorts, jeans, and a skirt. if a top and a bottom are chosen, make a list, a table, or a tree diagram to represent the sample space. 2. dee played a handheld game. there were 15 questions, of which she answered 8 incorrectly. at this rate, how many questions should dee expect to answer correctly if she answers a total of 135 questions? 3. in a deck of playing cards, what is the probability of drawing a red card, replacing it, and then drawing a queen?

Explanation:

1.

Step1: List all combinations

Let tops be Red (R), Green (G), Purple (P), Black (B) and bottoms be Shorts (S), Jeans (J), Skirt (K).
The sample - space is: (R,S), (R,J), (R,K), (G,S), (G,J), (G,K), (P,S), (P,J), (P,K), (B,S), (B,J), (B,K)

2.

Step1: Calculate the number of correct answers out of 15 questions

The number of correct answers out of 15 questions is \(15 - 8=7\).

Step2: Set up a proportion

Let \(x\) be the number of correct answers out of 135 questions. The proportion is \(\frac{7}{15}=\frac{x}{135}\).

Step3: Cross - multiply and solve for \(x\)

Cross - multiplying gives \(15x = 7\times135\), so \(x=\frac{7\times135}{15}\).
\(x = 63\)

3.

Step1: Calculate the probability of drawing a red card

A standard deck of 52 playing cards has 26 red cards. So the probability of drawing a red card \(P(R)=\frac{26}{52}=\frac{1}{2}\).

Step2: Calculate the probability of drawing a queen

A standard deck has 4 queens. So the probability of drawing a queen \(P(Q)=\frac{4}{52}=\frac{1}{13}\).

Step3: Calculate the probability of the two independent events

Since the events are independent (because the card is replaced), the probability of drawing a red card and then a queen is \(P = P(R)\times P(Q)\).
\(P=\frac{1}{2}\times\frac{1}{13}=\frac{1}{26}\)

Answer:

1. Sample - space: (R,S), (R,J), (R,K), (G,S), (G,J), (G,K), (P,S), (P,J), (P,K), (B,S), (B,J), (B,K)
2. 63
3. \(\frac{1}{26}\)