3.2 Roulette wheel. A "new and improved" version of roulette involves spinning a wheel with 23 red, 23 black, and 5 green slots. A ball is spun onto the wheel and will eventually land in a slot, where each slot has an equal chance of capturing the ball. (a) You watch a roulette wheel spin 3 consecutive times and the ball lands on a red slot each time. What is the probability that the ball will land on a red slot on the next spin?

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To solve this problem, we need to determine the probability of the ball landing on a red slot on the next spin of the roulette wheel, given the conditions of the game.

### Step 1: Identify the total number of slots
The roulette wheel has:
- 23 red slots
- 23 black slots
- 5 green slots

The total number of slots is:
$$
T = 23 + 23 + 5 = 51
$$

### Step 2: Calculate the probability of landing on a red slot
The probability $$ P(R) $$ of landing on a red slot in a single spin is given by the ratio of the number of red slots to the total number of slots:
$$
P(R) = \frac{\text{Number of red slots}}{\text{Total number of slots}} = \frac{23}{51}
$$

### Step 3: Consider the independence of spins
The spins of the roulette wheel are independent events. This means that the outcome of previous spins does not affect the outcome of the next spin. Therefore, the fact that the ball landed on red in the previous three spins does not change the probability of landing on red in the next spin.

### Conclusion
The probability that the ball will land on a red slot on the next spin remains:
$$
P(R) = \frac{23}{51}
$$

Thus, the final answer is:
$$
\frac{23}{51}
$$
The probability of the ball landing on a red slot on the next spin is $$ \frac{23}{51} $$.

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