To solve the problem step by step, we will first determine the total number of ways to choose 5 elements from the first 15 natural numbers, then calculate the number of ways to choose 5 elements such that no two elements are consecutive, and finally compute the probability.
### Step 1: Calculate the total number of ways to choose 5 elements from 15.
The total number of ways to choose 5 elements from a set of 15 is given by the combination formula:
\[
\text{Total cases} = \binom{15}{5}
\]
### Step 2: Calculate the number of ways to choose 5 elements such that no two are consecutive.
To ensure that no two chosen numbers are consecutive, we can think of placing "gaps" between the chosen numbers. If we choose 5 numbers, we need to leave at least 1 number between each pair of chosen numbers to avoid them being consecutive.
If we denote the chosen numbers as \( x_1, x_2, x_3, x_4, x_5 \), we can transform the problem by defining new variables:
\[
y_1 = x_1, \quad y_2 = x_2 - 1, \quad y_3 = x_3 - 2, \quad y_4 = x_4 - 3, \quad y_5 = x_5 - 4
\]
This transformation ensures that \( y_1, y_2, y_3, y_4, y_5 \) are all distinct and chosen from the set of numbers from 1 to \( 15 - 4 = 11 \) (since we have effectively removed 4 positions for the gaps).
Thus, the number of ways to choose 5 elements such that no two are consecutive is:
\[
\text{Unfavorable cases} = \binom{11}{5}
\]
### Step 3: Calculate the probability that at least two of the five numbers are consecutive.
The probability that at least two of the selected numbers are consecutive can be found using the complement rule:
\[
P(\text{at least 2 consecutive}) = 1 - P(\text{no two consecutive})
\]
Where:
\[
P(\text{no two consecutive}) = \frac{\text{Unfavorable cases}}{\text{Total cases}} = \frac{\binom{11}{5}}{\binom{15}{5}}
\]
Thus,
\[
P(\text{at least 2 consecutive}) = 1 - \frac{\binom{11}{5}}{\binom{15}{5}}
\]
### Step 4: Calculate the values of the combinations.
Calculating the combinations:
\[
\binom{15}{5} = \frac{15 \times 14 \times 13 \times 12 \times 11}{5 \times 4 \times 3 \times 2 \times 1} = 3003
\]
\[
\binom{11}{5} = \frac{11 \times 10 \times 9 \times 8 \times 7}{5 \times 4 \times 3 \times 2 \times 1} = 462
\]
### Step 5: Substitute the values into the probability formula.
Now substituting the values into the probability formula:
\[
P(\text{at least 2 consecutive}) = 1 - \frac{462}{3003}
\]
Calculating this gives:
\[
P(\text{at least 2 consecutive}) = 1 - 0.153 = 0.847
\]
### Step 6: Express the probability in terms of \( \lambda \).
Let \( \lambda = P(\text{at least 2 consecutive}) = \frac{3003 - 462}{3003} = \frac{2541}{3003} \).
### Step 7: Find the value of \( \frac{22}{\lambda} \).
Now we need to find:
\[
\frac{22}{\lambda} = \frac{22 \times 3003}{2541}
\]
Calculating this gives:
\[
\frac{22 \times 3003}{2541} = 26
\]
### Final Answer:
Thus, the value of \( \frac{22}{\lambda} \) is \( 26 \).
