$$ 10(5.1,5.2) $$ Question 24, 5.1.27 Part 5 of 5 HW Score: $$ 49.6 \%, 24.8 $$ of 50 points In a certain game of chance, a wheel consists of 48 slots numbered $$ 00,0,1,2, \ldots, 46 $$. To play the game, a metal ball is spun around the wheel and is allowed to fall into one of the numbered slots. Complete parts (a) through (c) below. (a) Determine the sample space. Choose the correct answer below. A. The sample space is $$ \{00,0,1,2, \ldots, 46\} $$. 8. The sample space is $$ \{1,2, \ldots, 46\} $$. C. The sample space is $$ \{00\} $$. D. The sample space is $$ \{00,0\} $$. (b) Determine the probability that the metal ball falls into the slot marked 4. Interpret this probability. The probability that the metal ball falls into the slot marked 4 is (Round to four decimal places as needed.) Interpret this probability. Select the correct choice below and fill in the answer box within your choice. (Type a whole number.) A. If the wheel is spun 1000 times, it is expected that about 21 of those times result in the ball landing in slot 4 . B. If the wheel is spun 1000 times, it is expected that exactly $$ \square $$ of those times result in the ball not landing in slot 4 . (c) Determine the probability that the metal ball lands in an odd slot. Interpret this probability. The probability that the metal ball lands in an odd slot is 0.4792 . (Round to four decimal places as needed.) Interpret this probability. Select the correct choice below and fill in the answer box within your choice. (Type a whole number.) A. If the wheel is spun 100 times, it is expected that about $$ \square $$ of those times result in the ball landing on an odd number. B. If the wheel is spun 100 times, it is expected that exactly $$ \square $$ mple Get more help .

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### Part (a): Determine the Sample Space

The sample space consists of all possible outcomes of the game. The wheel has slots numbered from $$00$$ to $$46$$, which includes:

- $$00$$
- $$0$$
- $$1$$
- $$2$$
- ...
- $$46$$

Thus, the sample space is:

$$
\{00, 0, 1, 2, \ldots, 46\}
$$

So the correct answer is:
**A. The sample space is $$ \{00,0,1,2, \ldots, 46\} $$.**

### Part (b): Probability of Landing in Slot Marked 4

To find the probability that the metal ball falls into the slot marked $$4$$, we can 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 (landing in slot $$4$$) is $$1$$.
- The total number of outcomes (the total slots) is $$48$$.

Thus, the probability is:

$$
P(4) = \frac{1}{48}
$$

Calculating this gives:

$$
P(4) \approx 0.0208
$$

Rounding to four decimal places, we have:

$$
P(4) \approx 0.0208
$$

**Interpretation:**
If the wheel is spun $$1000$$ times, we can expect the ball to land in slot $$4$$ approximately:

$$
1000 \times 0.0208 \approx 20.8333
$$

Rounding this to a whole number gives us $$21$$.

So the correct interpretation is:
**A. If the wheel is spun 1000 times, it is expected that about 21 of those times result in the ball landing in slot 4.**

### Part (c): Probability of Landing in an Odd Slot

To find the probability that the metal ball lands in an odd slot, we first identify the odd slots in the sample space:

The odd slots are:
- $$1, 3, 5, 7, \ldots, 45$$

The odd numbers from $$1$$ to $$46$$ can be calculated as follows:
- The sequence of odd numbers is an arithmetic sequence where the first term $$a = 1$$ and the common difference $$d = 2$$.

To find the number of odd slots, we can use the formula for the $$n$$-th term of an arithmetic sequence:

$$
a_n = a + (n-1)d
$$

Setting $$a_n = 45$$:

$$
45 = 1 + (n-1) \cdot 2
$$
$$
44 = (n-1) \cdot 2
$$
$$
n-1 = 22
$$
$$
n = 23
$$

So there are $$23$$ odd slots.

Now, we can calculate the probability:

$$
P(\text{odd}) = \frac{\text{Number of odd slots}}{\text{Total number of slots}} = \frac{23}{48}
$$

Calculating this gives:

$$
P(\text{odd}) \approx 0.4792
$$

**Interpretation:**
If the wheel is spun $$100$$ times, we can expect the ball to land in an odd slot approximately:

$$
100 \times 0.4792 \approx 47.92
$$

Rounding this to a whole number gives us $$48$$.

So the correct interpretation is:
**A. If the wheel is spun 100 times, it is expected that about 48 of those times result in the ball landing on an odd number.**

### Summary of Answers:
- (a) A. The sample space is $$ \{00,0,1,2, \ldots, 46\} $$.
- (b) Probability: $$0.0208$$; Interpretation: A. About 21 times.
- (c) Probability: $$0.4792$$; Interpretation: A. About 48 times.
**Part (a):** The sample space is $$ \{00, 0, 1, 2, \ldots, 46\} $$. **Part (b):** The probability of landing in slot 4 is $$0.0208$$. If the wheel is spun 1000 times, it is expected that about 21 times the ball lands in slot 4. **Part (c):** The probability of landing in an odd slot is $$0.4792$$. If the wheel is spun 100 times, it is expected that about 48 times the ball lands on an odd number.

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