The time taken to complete a motorcycle race is normally distributed, with an average time ( $$ \mu $$ ) of 2.5 hours and a standard deviation $$ (\sigma) $$ of 0.5 hours. What is the probability that a randomly selected cyclist will take between 2.35 and 2.45 hours to complete the race?

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To find the probability that a randomly selected cyclist will take between 2.35 and 2.45 hours to complete the race, we can use the properties of the normal distribution.

### Step 1: Standardize the values
We will convert the times into z-scores using the formula:

$$
z = \frac{x - \mu}{\sigma}
$$

where:
- $$ x $$ is the value we are standardizing,
- $$ \mu = 2.5 $$ hours (mean),
- $$ \sigma = 0.5 $$ hours (standard deviation).

### Step 2: Calculate the z-scores
1. For $$ x = 2.35 $$:
   $$
   z_1 = \frac{2.35 - 2.5}{0.5}
   $$
2. For $$ x = 2.45 $$:
   $$
   z_2 = \frac{2.45 - 2.5}{0.5}
   $$

Let's calculate these z-scores.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{\left(2.45-2.5\right)}{0.5}$$
- step1: Remove the parentheses:
  $$\frac{2.45-2.5}{0.5}$$
- step2: Subtract the numbers:
  $$\frac{-0.05}{0.5}$$
- step3: Convert the expressions:
  $$\frac{-\frac{1}{20}}{\frac{1}{2}}$$
- step4: Multiply by the reciprocal:
  $$-\frac{1}{20}\times 2$$
- step5: Reduce the numbers:
  $$-\frac{1}{10}\times 1$$
- step6: Multiply:
  $$-\frac{1}{10}$$
Calculate or simplify the expression $$ (2.35 - 2.5) / 0.5 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{\left(2.35-2.5\right)}{0.5}$$
- step1: Remove the parentheses:
  $$\frac{2.35-2.5}{0.5}$$
- step2: Subtract the numbers:
  $$\frac{-0.15}{0.5}$$
- step3: Convert the expressions:
  $$\frac{-\frac{3}{20}}{\frac{1}{2}}$$
- step4: Multiply by the reciprocal:
  $$-\frac{3}{20}\times 2$$
- step5: Reduce the numbers:
  $$-\frac{3}{10}\times 1$$
- step6: Multiply:
  $$-\frac{3}{10}$$
It seems that I cannot directly calculate the probabilities for the z-scores using the current tools. However, I can provide the general approach to find these probabilities.

### Step 4: Use the standard normal distribution
You can use a standard normal distribution table or a calculator to find the probabilities:

1. For $$ z_1 = -0.3 $$:
   - Look up the value in the z-table or use a calculator to find $$ P(Z < -0.3) $$.

2. For $$ z_2 = -0.1 $$:
   - Look up the value in the z-table or use a calculator to find $$ P(Z < -0.1) $$.

### Step 5: Calculate the probability between the two z-scores
The probability that a randomly selected cyclist will take between 2.35 and 2.45 hours is given by:

$$
P(2.35 < X < 2.45) = P(Z < z_2) - P(Z < z_1)
$$

### Example Values
Using standard normal distribution tables:
- $$ P(Z < -0.3) \approx 0.3821 $$
- $$ P(Z < -0.1) \approx 0.4602 $$

### Final Calculation
Now, we can calculate the probability:

$$
P(2.35 < X < 2.45) = P(Z < -0.1) - P(Z < -0.3) \approx 0.4602 - 0.3821
$$

Let's calculate this final probability.
Calculate the value by following steps:
- step0: Calculate:
  $$0.4602-0.3821$$
- step1: Subtract the numbers:
  $$0.0781$$
The probability that a randomly selected cyclist will take between 2.35 and 2.45 hours to complete the race is approximately:

$$
P(2.35 < X < 2.45) \approx 0.0781
$$

### Conclusion
Thus, the probability is about **0.0781**, or **7.81%**.
The probability that a randomly selected cyclist will take between 2.35 and 2.45 hours to complete the race is approximately 7.81%.

The time taken to complete a motorcycle race is normally distributed, with an average time ( \( \mu \) ) of 2.5 hours and a standard deviation \( (\sigma) \) of 0.5 hours. What is the probability that a randomly selected cyclist will take between 2.35 and 2.45 hours to complete the race?

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