Sure, let's go through these questions step-by-step.
### 6. Estimating the Probability from the Given Data
(a) The drawing pin is thrown 10 times, and it lands point up 6 times. To estimate the probability that it lands point up from this data, we use the formula for empirical probability:
[tex]\[ P(U) = \frac{\text{Number of times it lands point up}}{\text{Total number of throws}} \][/tex]
From the given data:
[tex]\[ P(U) = \frac{6}{10} = 0.6 \][/tex]
(b) Now, suppose the drawing pin is thrown 500 times and lands point up 285 times. Again, we estimate the probability using the same empirical probability formula:
[tex]\[ P(U) = \frac{\text{Number of times it lands point up}}{\text{Total number of throws}} \][/tex]
From the given data:
[tex]\[ P(U) = \frac{285}{500} = 0.57 \][/tex]
(c) To determine which probability estimate is more reliable, we consider the number of trials. Generally, estimates from a larger number of trials tend to be more reliable because they are less likely to be affected by random fluctuations. So, the more reliable estimate would be the one with 500 trials:
[tex]\[ \text{The more reliable estimate is the one with 500 trials, i.e., } 0.57. \][/tex]
### 7. Why Must a Probability Value, [tex]\( P(E) \)[/tex], Be in the Range [tex]\( 0 \leq P(E) \leq 1 \)[/tex]?
Probability values represent the likelihood of an event occurring. By definition, the probability of an event [tex]\( E \)[/tex] must satisfy the following conditions:
- [tex]\( P(E) = 0 \)[/tex] means the event [tex]\( E \)[/tex] will not occur under any circumstances.
- [tex]\( P(E) = 1 \)[/tex] means the event [tex]\( E \)[/tex] is certain to occur.
- For any event [tex]\( E \)[/tex], the probability [tex]\( P(E) \)[/tex] must be somewhere between these two extremes, inclusive.
This is why probabilities are constrained within the range:
[tex]\[ 0 \leq P(E) \leq 1 \][/tex]
In essence, any probability value outside this range would not make sense in the context of likelihood since:
- A value less than 0 would imply a negative likelihood, which is not possible.
- A value greater than 1 would imply a likelihood greater than certainty, which is also not possible.
Thus, probabilities are always within the range of 0 and 1, inclusive.
