ANSWER:

The probability of not getting the character in one turn is 1 - 0.6\% = 99.4%

To find the number of turns required for a guaranteed success, set up the inequality:

(0.994)^n < 0.5

Now, solve for (n). The answer will be the smallest integer greater than or equal to the solution.

The number of turns required for a guaranteed success is approximately 1158.

STEP BY STEP EXPLANATION

(0.994)^n < 0.5

Taking the natural logarithm of both sides:

\[ \ln(0.994)^n < \ln(0.5) \]

Using the property \( \ln(a^b) = b \cdot \ln(a) \):

\[ n \cdot \ln(0.994) < \ln(0.5) \]

Now, solve for \(n\):

\[ n > \frac{\ln(0.5)}{\ln(0.994)} \]

\[ n > \frac{-0.6931}{-0.006}\]

\[ n > 1157.86 \]

So, you would need at least 1158 turns for a guaranteed chance of getting the character in the worst-case scenario.

Answer:

To calculate the number of turns required to guarantee getting the character in the worst-case scenario, we can use the concept of the geometric distribution. In this case, the probability of getting the character in a single turn is 0.6% (or 0.006 as a decimal).

The geometric distribution represents the number of trials needed to achieve the first success (in this case, obtaining the character) with a given probability. The formula to calculate the expected number of trials (E[X]) is:

E[X] = 1 / p

where p is the probability of success in a single trial.

So, in this case, the expected number of turns required to guarantee getting the character would be:

E[X] = 1 / 0.006 = 166.67

Since we can't have a fraction of a turn, we would need to round up to the nearest whole number. Therefore, in the worst-case scenario, it would take at least 167 turns to guarantee getting the character.