Understanding the probability of events is crucial in many scenarios, including sports like ten-pin bowling. This question asks us to determine the minimum number of times a bowler must roll the ball to achieve a certain probability of hitting the target at least once.

Probability Basics for Bowling

First, let's define the given probabilities involved in the bowling scenario:

• The probability of the bowler hitting the target (which we consider a success), denoted as \(P(\text{Hit})\), is given as \(\frac{3}{4}\).
• The probability of the bowler not hitting the target (which is a failure), denoted as \(P(\text{No Hit})\), is the complement of hitting the target.

We can calculate \(P(\text{No Hit})\) as follows:

\[P(\text{No Hit}) = 1 - P(\text{Hit})\]

Substituting the given value:

\[P(\text{No Hit}) = 1 - \frac{3}{4}\]

\[P(\text{No Hit}) = \frac{1}{4}\]

Minimum Rolls Calculation

We are looking for the minimum number of rolls, let's call this \(n\), such that the probability of hitting the target at least once is more than 0.99.

The event "hitting the target at least once" is the opposite of the event "not hitting the target at all" (meaning the bowler misses the target on every single roll).

So, we can express the probability of "at least one hit" using its complement:

\[P(\text{At least one hit}) = 1 - P(\text{No hits in } n \text{ rolls})\]

Since each roll is an independent event, the probability of not hitting the target in \(n\) consecutive rolls is the product of the probabilities of not hitting the target for each roll:

\[P(\text{No hits in } n \text{ rolls}) = \left(P(\text{No Hit})\right)^n\]

\[P(\text{No hits in } n \text{ rolls}) = \left(\frac{1}{4}\right)^n\]

Now, we can set up the inequality based on the condition given in the problem statement:

\[P(\text{At least one hit}) > 0.99\]

Substitute the expression we found:

\[1 - \left(\frac{1}{4}\right)^n > 0.99\]

Solving the Inequality for Rolls

Let's rearrange this inequality to solve for \(n\):

First, subtract 1 from both sides of the inequality:

\[-\left(\frac{1}{4}\right)^n > 0.99 - 1\]

\[-\left(\frac{1}{4}\right)^n > -0.01\]

Next, multiply both sides by -1. Remember that when you multiply or divide an inequality by a negative number, you must reverse the inequality sign:

\[\left(\frac{1}{4}\right)^n < 0.01\]

We can also write 0.01 as a fraction, \(\frac{1}{100}\):

\[\left(\frac{1}{4}\right)^n < \frac{1}{100}\]

This means:

\[\frac{1}{4^n} < \frac{1}{100}\]

To compare the values more easily, we can take the reciprocal of both sides. This also requires reversing the inequality sign:

\[4^n > 100\]

Evaluating Number of Rolls

Now, we need to find the smallest integer value for \(n\) that satisfies the condition \(4^n > 100\). Let's test values of \(n\) starting from 1:

• If \(n = 1\): Calculate \(4^1 = 4\). Since \(4 \not> 100\), \(n=1\) is not enough.
• If \(n = 2\): Calculate \(4^2 = 16\). Since \(16 \not> 100\), \(n=2\) is not enough.
• If \(n = 3\): Calculate \(4^3 = 64\). Since \(64 \not> 100\), \(n=3\) is not enough.
• If \(n = 4\): Calculate \(4^4 = 256\). Since \(256 > 100\), \(n=4\) satisfies the condition.

Therefore, the minimum number of times the bowler must roll down so that the probability of hitting the target at least once is more than 0.99 is 4 times.