8. Solution. Let the correct numbers in the table be $a, b, c$ and $d$. Consider the events $A=\{$ Has a card $\}$ and $B=\{$ Makes online purchases $\}$. These events are independent if the proportion of cardholders among online shoppers is the same as among those who do not make online purchases, that is,

|  | Has a card | No card |
| :--- | :---: | :---: |
| Buys online | $a$ | $b$ |
| Does not buy online | $c$ | $d$ |

the conditional probability of event $A$ given $B$ is equal to the probability of event $A$ given $\bar{B}: \mathrm{P}(A \mid B)=\mathrm{P}(A \mid \bar{B})$. We get $\frac{a}{a+b}=\frac{c}{c+d}$, from which $a d=b c$.

Due to random variability in the data, this equality may not be exact, but it should at least hold approximately. Suppose that all numbers in the Scientist's report are correct except for $a$. Then the number $a$ should approximately (rounded to the nearest integer) equal $\frac{b c}{d}=\frac{245 \cdot 1142}{535} \approx 523$, and the number of respondents will be approximately $523+1142+535+245=2445$. This is much greater than 2000. The assumption is implausible.

Suppose now that the incorrect number is $b$. Then $b \approx \frac{a d}{c}=\frac{81 \cdot 535}{1142} \approx 38$, and the total number of survey participants is close to $1142+535+81+38=1796$. This is possible.

If the incorrect number is $c$, then $c \approx \frac{a d}{b}=\frac{81 \cdot 535}{245} \approx 177$, and the approximate total number of respondents is $177+535+81+245=1038$. This is too small: according to the condition, the sample size is more than 1500 people.

If the incorrect number is $d$, then $d \approx \frac{c b}{a}=\frac{1142 \cdot 245}{81} \approx 3454$, and the total sample size is approximately $1142+245+81+3454=4922$. This is also implausible.
Answer: approximately 1796 people.

Note. Instead of approximate equalities, estimates can be made using inequalities.

## Grading Criteria

| Solution is complete and correct | 2 points |
| :--- | :--- |
| Answer is correct, but there are no arguments showing that other options are implausible (for example, there is a guess about which number is incorrect) | 1 point |
| Solution is incorrect or missing | 0 points |