High School

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Error Analysis: A student randomly draws a whole number between 1 and 30. Describe and correct the error in finding the probability that the number drawn is greater than 4.

**Given:**

\[
P(\text{number} \ \textgreater \ 4) = 1 - P(\text{number} \ \leq \ 4)
\]

**Calculation:**

\[
P(\text{number} \ \textless \ 4) = \frac{3}{30}
\]

\[
P(\text{number} \ \textgreater \ 4) = 1 - \frac{3}{30} = \frac{27}{30} = \frac{9}{10}
\]

**Correction:**

The error is in the calculation of \( P(\text{number} \ \leq \ 4) \). Since numbers less than or equal to 4 are 1, 2, 3, and 4, the probability should be calculated as:

\[
P(\text{number} \ \leq \ 4) = \frac{4}{30} = \frac{2}{15}
\]

Thus, the correct calculation for \( P(\text{number} \ \textgreater \ 4) \) should be:

\[
P(\text{number} \ \textgreater \ 4) = 1 - \frac{4}{30} = \frac{26}{30} = \frac{13}{15}
\]

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]

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Rewritten by : Barada