IBDP Maths analysis and approaches Topic: SL 4.3- Measures of central tendency (mean, median and mode) HL Paper 1

(a) (i) To estimate the probability that a randomly selected visitor will ride \textit{The Dragon} on Tuesday, we consider the frequency data provided from Monday’s sample. The total number of visitors sampled is 40. The number of visitors who rode \textit{The Dragon} at least once is the sum of the frequencies for one or more rides. This can be calculated as follows:
16 visitors rode \textit{The Dragon} once,
13 visitors rode it twice,
2 visitors rode it three times, and
3 visitors rode it four times.
Thus, the total number of visitors who rode \textit{The Dragon} is:
\(
16 + 13 + 2 + 3 = 34
\)
The probability that a randomly selected visitor will ride \textit{The Dragon} is therefore the number of visitors who rode the ride divided by the total number of visitors sampled. Hence, the probability is:
\(
\frac{34}{40}
\)
which simplifies to:
\(
\frac{17}{20}
\)
or \( 0.85 \) when expressed as a decimal.
Therefore, the estimated probability for a randomly selected visitor to ride \textit{The Dragon} on Tuesday is:
\(
0.85
\)

      (ii) The expected number of times a visitor will ride \textit{The Dragon} is calculated using the expected value formula in probability, which is the sum of the products of the values and their corresponding probabilities:
\(
\textbf{Expected Value (}E(X)\textbf{)} = \sum [x \times P(X = x)]
\)
To find the expected value, we first determine the probability of each outcome by dividing the frequency of each outcome by the total number of responses (40 in this case). Then, each outcome is multiplied by its probability:
\(
P(0 \text{ rides}) = \frac{6}{40}
\)
\(
P(1 \text{ ride}) = \frac{16}{40}
\)
\(
P(2 \text{ rides}) = \frac{13}{40}
\)
\(
P(3 \text{ rides}) = \frac{2}{40}
\)
\(
P(4 \text{ rides}) = \frac{3}{40}
\)
These probabilities are then used to calculate the expected value:
\(
E(X) = (0 \times \frac{6}{40}) + (1 \times \frac{16}{40}) + (2 \times \frac{13}{40}) + (3 \times \frac{2}{40}) + (4 \times \frac{3}{40})
\)
\(
E(X) = \left(0\right) + \left(\frac{16}{40}\right) + \left(\frac{26}{40}\right) + \left(\frac{6}{40}\right) + \left(\frac{12}{40}\right)
\)
\(
E(X) = \frac{60}{40}
\)
\(
E(X) = 1.5
\)
Therefore, the expected number of times a visitor will ride \textit{The Dragon} is \textbf{1.5 times}.

(b) The first step in determining the minimum number of times The Dragon must run to satisfy demand is to calculate the expected number of rides per visitor. This is achieved by finding the weighted average of rides based on the provided frequency data. The process involves multiplying the number of times on The Dragon by the corresponding frequency and then summing these products. The sum is then divided by the total number of sampled visitors, which yields the average number of rides per visitor.
For the sample data provided, the calculation is as follows:
\(
(0 \text{ rides} \times 6 \text{ visitors}) + (1 \text{ ride} \times 16 \text{ visitors}) + (2 \text{ rides} \times 13 \text{ visitors}) + (3 \text{ rides} \times 2 \text{ visitors}) + (4 \text{ rides} \times 3 \text{ visitors})
\)
\(
= 0 + 16 + 26 + 6 + 12 = 60 \text{ rides}
\)
\(
\frac{60 \text{ rides}}{40 \text{ visitors}} = 1.5 \text{ rides per visitor}
\)
Given that 1000 visitors are expected on Tuesday, the total number of rides needed is:
\(
\frac{1.5 \text{ rides}}{\text{visitor}} \times 1000 \text{ visitors} = 1500 \text{ rides}
\)
The Dragon has a maximum capacity of 10 people per run, therefore, the minimum number of runs required to meet the total demand of rides is:
\(
\frac{1500 \text{ rides}}{10 \text{ people per run}} = 150 \text{ runs}
\)
Therefore, to satisfy the demand, The Dragon must run a minimum of 150 times.