Question
In a certain game of chance, a wheel consists of 34 slots numbered \( 00,0,1,2 \ldots, 32 \). To play the game, a metal ball is spun around the wheel and is allowe parts (a) through (c) below. (a) Determine the sample space. Choose the correct answer below. A. The sample space is \( \{00\} \). in The sample space is \( \{1,2,32\} \) C. The sample space is \( \{00,0\} \). D. The sample space is \( \langle 00,0,1,2, \ldots, 32\} \). (b) Delermine the probability that the metal ball falls into the slot marked 7. Interpret this probability. The probability that the metal ball falls into the slot marked 7 is \( \square \) (Round to lour decimal places as needed.)
Ask by Salinas Cervantes. in the United States
Feb 27,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
**Part (a):** The sample space is \( \{00, 0, 1, 2, \ldots, 32\} \).
**Part (b):** The probability that the metal ball falls into the slot marked 7 is \( 0.0294 \).
Solution
Let's solve the problem step by step.
### Part (a): Determine the Sample Space
The sample space is the set of all possible outcomes of an experiment. In this case, the experiment is spinning the wheel, which has 34 slots numbered from \( 00, 0, 1, 2, \ldots, 32 \).
The correct representation of the sample space includes all these numbers. Therefore, the sample space can be expressed as:
\[
S = \{00, 0, 1, 2, \ldots, 32\}
\]
Looking at the options provided:
- A. The sample space is \( \{00\} \). (Incorrect)
- B. The sample space is \( \{1,2,32\} \). (Incorrect)
- C. The sample space is \( \{00,0\} \). (Incorrect)
- D. The sample space is \( \{00,0,1,2,\ldots,32\} \). (Correct)
Thus, the answer for part (a) is:
**D. The sample space is \( \{00,0,1,2,\ldots,32\} \).**
### Part (b): Determine the Probability of Falling into Slot Marked 7
To find the probability that the metal ball falls into the slot marked 7, we use the formula for probability:
\[
P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}
\]
In this case:
- The number of favorable outcomes (falling into slot 7) is 1.
- The total number of outcomes (total slots) is 34.
Now we can calculate the probability:
\[
P(7) = \frac{1}{34}
\]
Next, we will round this probability to four decimal places:
\[
P(7) \approx 0.0294
\]
Thus, the probability that the metal ball falls into the slot marked 7 is:
**The probability that the metal ball falls into the slot marked 7 is \( 0.0294 \).**
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Simplify this solution Bonus Knowledge
The sample space of the game includes all the slots the ball can land in, which are from \(00\) to \(32\). Therefore, the correct answer is \( D. \) The sample space is \( \{00, 0, 1, 2, \ldots, 32\} \). To find the probability of the metal ball landing in the slot marked 7, we consider there are 34 equally likely slots. The probability \( P \) of landing in slot \(7\) is given by the formula \(P = \frac{1}{\text{total number of slots}}\). Thus: \[ P(7) = \frac{1}{34} \approx 0.0294 \] This means there’s about a 2.94% chance of the ball landing in the slot marked 7.
