Answer :
Consider the following step-by-step solution:
1. There are [tex]$30$[/tex] equally likely whole numbers (from [tex]$1$[/tex] to [tex]$30$[/tex]) in the sample space.
2. We are interested in finding the probability that a number drawn is greater than [tex]$4$[/tex]. A standard method to compute this probability is to subtract the probability of the complement event from [tex]$1$[/tex].
3. The complement of the event “number [tex]$>4$[/tex]” is “number [tex]$\le 4$[/tex].” This means we have to count all numbers that are less than or equal to [tex]$4$[/tex].
4. The numbers less than or equal to [tex]$4$[/tex] are: [tex]$1$[/tex], [tex]$2$[/tex], [tex]$3$[/tex], and [tex]$4$[/tex]. There are [tex]$4$[/tex] such numbers.
5. Therefore, the probability of drawing a number that is [tex]$\le 4$[/tex] is
[tex]$$P(\text{number} \le 4) = \frac{4}{30}.$$[/tex]
6. Using the complement rule, the probability of drawing a number greater than [tex]$4$[/tex] is
[tex]$$P(\text{number} > 4) = 1 - P(\text{number} \le 4) = 1 - \frac{4}{30} = \frac{26}{30}.$$[/tex]
7. The fraction [tex]$\frac{26}{30}$[/tex] can be simplified by dividing the numerator and denominator by [tex]$2$[/tex], which gives
[tex]$$P(\text{number} > 4) = \frac{13}{15}.$$[/tex]
8. The error in the original approach was that the student subtracted the probability of numbers less than [tex]$4$[/tex] (i.e., [tex]$1$[/tex], [tex]$2$[/tex], and [tex]$3$[/tex]) instead of numbers less than or equal to [tex]$4$[/tex]. This led to
[tex]$$1 - \frac{3}{30} = \frac{27}{30} = \frac{9}{10},$$[/tex]
which is incorrect because it fails to include the number [tex]$4$[/tex] in the complement.
Thus, the correct probability that a randomly drawn number from [tex]$1$[/tex] to [tex]$30$[/tex] is greater than [tex]$4$[/tex] is
[tex]$$\boxed{\frac{13}{15}}.$$[/tex]
1. There are [tex]$30$[/tex] equally likely whole numbers (from [tex]$1$[/tex] to [tex]$30$[/tex]) in the sample space.
2. We are interested in finding the probability that a number drawn is greater than [tex]$4$[/tex]. A standard method to compute this probability is to subtract the probability of the complement event from [tex]$1$[/tex].
3. The complement of the event “number [tex]$>4$[/tex]” is “number [tex]$\le 4$[/tex].” This means we have to count all numbers that are less than or equal to [tex]$4$[/tex].
4. The numbers less than or equal to [tex]$4$[/tex] are: [tex]$1$[/tex], [tex]$2$[/tex], [tex]$3$[/tex], and [tex]$4$[/tex]. There are [tex]$4$[/tex] such numbers.
5. Therefore, the probability of drawing a number that is [tex]$\le 4$[/tex] is
[tex]$$P(\text{number} \le 4) = \frac{4}{30}.$$[/tex]
6. Using the complement rule, the probability of drawing a number greater than [tex]$4$[/tex] is
[tex]$$P(\text{number} > 4) = 1 - P(\text{number} \le 4) = 1 - \frac{4}{30} = \frac{26}{30}.$$[/tex]
7. The fraction [tex]$\frac{26}{30}$[/tex] can be simplified by dividing the numerator and denominator by [tex]$2$[/tex], which gives
[tex]$$P(\text{number} > 4) = \frac{13}{15}.$$[/tex]
8. The error in the original approach was that the student subtracted the probability of numbers less than [tex]$4$[/tex] (i.e., [tex]$1$[/tex], [tex]$2$[/tex], and [tex]$3$[/tex]) instead of numbers less than or equal to [tex]$4$[/tex]. This led to
[tex]$$1 - \frac{3}{30} = \frac{27}{30} = \frac{9}{10},$$[/tex]
which is incorrect because it fails to include the number [tex]$4$[/tex] in the complement.
Thus, the correct probability that a randomly drawn number from [tex]$1$[/tex] to [tex]$30$[/tex] is greater than [tex]$4$[/tex] is
[tex]$$\boxed{\frac{13}{15}}.$$[/tex]
