Question

evaluate independent practice lesson 5.1 homework
□ complete problems 3,5,7,9,10,13,14,17 for independent practice.
when you are finished, check the solutions with your teacher.
find the number of possible outcomes in the sample space. (see example 1)

1. you roll a die and flip three coins.

find the probability given the situation. (see example 2)

1. a game show airs on television five days per week, with a new player each day. each day, a prize is randomly placed behind one of two doors. the contestant wins the prize by selecting the correct door. what is the probability that exactly two of the five contestants win a prize during a week? hint: use the possible outcomes chart below.

possible outcomes

number of winners	outcomes
1	wllll, lwlll, llwll, lllwl, llllw
2	wwlll, wlwll, wllwl, wlllw, lwwll, lwlwl, lwllw, llwwl, llwlw, lllww
3	wwwll, wwlwl, wwllw, wlwwl, wlwlw, wllww, lwww l, lwwlw, lwlww, llwww
4	wwwwl, wwwlw, wwlww, wlwww, lwwww
5	wwwww

Explanation:

Problem 3:

Step1: Determine outcomes for die

A die has 6 faces, so outcomes for rolling a die: \( n_{\text{die}} = 6 \)

Step2: Determine outcomes for three coins

Each coin has 2 outcomes (heads/tails). For 3 coins, by multiplication principle: \( n_{\text{coins}} = 2\times2\times2 = 2^3 = 8 \)

Step3: Total outcomes in sample space

By multiplication principle (since die and coin flips are independent), total outcomes: \( n_{\text{total}} = n_{\text{die}} \times n_{\text{coins}} = 6\times8 = 48 \)

Step1: Find total number of possible outcomes

Each day has 2 outcomes (win/lose), for 5 days: \( n_{\text{total}} = 2^5 = 32 \)

Step2: Find number of favorable outcomes (exactly 2 winners)

From the table, count the outcomes for 2 winners:

• WWLLL, WLWL L, WLLWL, WLLLW, LWWLL, LWLWL, LWLLW, LLWWL, LLWLW, LLLWW

Total favorable outcomes: \( n_{\text{favorable}} = 10 \)

Step3: Calculate probability

Probability \( P = \frac{n_{\text{favorable}}}{n_{\text{total}}} = \frac{10}{32} = \frac{5}{16} \)

Answer:

48

Problem 7: