Certainly! Let's analyze the pattern in the probability of getting greater numbers of successes step by step.
1. Define the Probabilities:
- The probability of getting exactly 7 correct answers, [tex]\(P(7)\)[/tex], is 0.0031.
- The probability of getting exactly 8 correct answers, [tex]\(P(8)\)[/tex], is 0.000386.
- The probability of getting exactly 9 correct answers, [tex]\(P(9)\)[/tex], is 2.86 \times 10^{-5}.
- The probability of getting exactly 10 correct answers, [tex]\(P(10)\)[/tex], is 9.54 \times 10^{-7}.
2. Calculate the Ratios Between Successive Probabilities:
- The ratio of the probability of getting exactly 8 correct answers to the probability of getting exactly 7 correct answers is:
[tex]\[
\text{Ratio}_7^8 = \frac{P(8)}{P(7)} = \frac{0.000386}{0.0031} \approx 0.1245
\][/tex]
- The ratio of the probability of getting exactly 9 correct answers to the probability of getting exactly 8 correct answers is:
[tex]\[
\text{Ratio}_8^9 = \frac{P(9)}{P(8)} = \frac{2.86 \times 10^{-5}}{0.000386} \approx 0.0741
\][/tex]
- The ratio of the probability of getting exactly 10 correct answers to the probability of getting exactly 9 correct answers is:
[tex]\[
\text{Ratio}_9^{10} = \frac{P(10)}{P(9)} = \frac{9.54 \times 10^{-7}}{2.86 \times 10^{-5}} \approx 0.0334
\][/tex]
3. Calculate the Average Ratio:
- To find the average ratio of change in probability as the number of correct answers increases, sum the ratios calculated above and divide by the number of ratios:
[tex]\[
\text{Average Ratio} = \frac{\text{Ratio}_7^8 + \text{Ratio}_8^9 + \text{Ratio}_9^{10}}{3}
\][/tex]
- Plugging in the values we found:
[tex]\[
\text{Average Ratio} = \frac{0.1245 + 0.0741 + 0.0334}{3} \approx 0.0773
\][/tex]
Summary:
- The ratio [tex]\( \frac{P(8)}{P(7)} \)[/tex] is approximately 0.1245.
- The ratio [tex]\( \frac{P(9)}{P(8)} \)[/tex] is approximately 0.0741.
- The ratio [tex]\( \frac{P(10)}{P(9)} \)[/tex] is approximately 0.0334.
- The average ratio of change in probability as the number of correct answers increases is approximately 0.0773.
This indicates that the probability of getting a higher number of correct answers decreases significantly as the number of successes increases, and this decrease can be observed through the average ratio of roughly 0.0773.
