Question

cl 1-148. the county-fair prize wheel has equally spaced sections with the following colors: one is golden, two are silver, three are green, four are blue, six are red, and nine are yellow. a. what is the probability of landing on gold? give your answer as a fraction and as a percent. b. if the probability of landing on yellow is 36%, what is the probability of not landing on yellow? c. if the wheel is spun 100 times, how many times would you expect to land on silver? core connections, course 2 chapter

Explanation:

Part a

Step1: Calculate total sections

First, we find the total number of sections on the wheel. We add up the number of each color: \(1 + 2 + 3 + 4 + 6 + 9\). Let's compute that: \(1+2 = 3\), \(3+3 = 6\), \(6+4 = 10\), \(10+6 = 16\), \(16+9 = 25\). So total sections \(n = 25\).

Step2: Probability of gold

The number of gold sections is \(1\). The probability \(P\) of an event is the number of favorable outcomes over total outcomes. So for gold, \(P(\text{gold})=\frac{\text{number of gold sections}}{\text{total sections}}=\frac{1}{25}\). To convert to a percent, we multiply by \(100\): \(\frac{1}{25}\times100 = 4\%\).

Step1: Use complement rule

The probability of an event and its complement (not the event) add up to \(1\) (or \(100\%\)). Let \(P(Y)\) be the probability of landing on yellow (\(36\% = 0.36\)) and \(P(\text{not }Y)\) be the probability of not landing on yellow. Then \(P(\text{not }Y)=1 - P(Y)\).

Step2: Calculate the probability

Substitute \(P(Y) = 0.36\) into the formula: \(1 - 0.36 = 0.64\), which is \(64\%\).

Step1: Probability of silver

First, find the probability of landing on silver. The number of silver sections is \(2\), total sections is \(25\) (from part a). So \(P(\text{silver})=\frac{2}{25}\).

Step2: Expected number of times

To find the expected number of times we land on silver in \(100\) spins, we multiply the probability by the number of trials (spins). So \(E = 100\times\frac{2}{25}\). Simplify: \(100\div25 = 4\), \(4\times2 = 8\).

Answer:

As a fraction: \(\frac{1}{25}\), as a percent: \(4\%\)

Part b