Probability

1103164505

Level: 
B
Suppose we have a rectangular fish tank which is \( 4\,\mathrm{dm} \) long, \( 2\,\mathrm{dm} \) wide, and it is filled with water up to the height of \( 3\,\mathrm{dm} \). In its four bottom corners, there are jets through which a fresh air is driven into the water in specific intervals. The fresh air is driven to the distance of up to \( 5\,\mathrm{cm} \) from the tank corners. If a fish swims inside the tank, what is the probability that the fish will not be hit by the stream of bubbles, at the moment when all four jets are acting? The fish dimensions can be neglected, round the result to \( 4 \) decimal places.
\( 0.9891 \)
\( 0.0109 \)
\( 0.9984 \)
\( 0.0016 \)
\( 0.9782 \)
\( 0.0218 \)

1103164504

Level: 
B
An equilateral triangle is drawn on the wall. Inside the triangle, there is a circle with the radius of \( 1 \) metre inscribed. If a fly sits by chance in the triangle, what is the probability that it does not sit inside the circle? Round the result to \( 4 \) decimal places.
\( 0.3954 \)
\( 0.6046 \)
\( 0.3023 \)
\( 0.6977 \)

1103164503

Level: 
B
An equilateral triangle with a side of 3 metres is drawn on the wall. Inside the triangle, there is a circle with the diameter of 1 metre. If a fly sits by chance in the triangle, what is the probability that it does not sit inside the circle? Round the result to 4 decimal places.
\( 0.7985 \)
\( 0.2015 \)
\( 0.8061 \)
\( 0.1939 \)

1003164502

Level: 
B
Suppose the points \( A \) and \( B \) are randomly placed on a circle with a radius \( r \). What is the probability that the distance between \( A \) and \( B \) (length of the chord \( AB \)) is at least \( r \)?
\( \frac23 \)
\( \frac13 \)
\( \frac16 \)
\( \frac56 \)
\( \frac12 \)

1003164501

Level: 
B
In the house with a \( 7 \) metre high ground storey and \( 6 \) other storeys (\( 5 \) metres high each), there is a lift. On each storey, it is possible to enter this lift through a glass door that is \( 2 \) metres high. The lift malfunctioned and stopped somewhere on its way. What is the probability that (at the moment of stoppage) it would not be possible to see out of the lift only the wall of the lift shaft?
\( 0.7500 \)
\( 0.7838 \)
\( 0.7188 \)
\( 0.7647 \)
\( 0.7353 \)
\( 0.7568 \)

1003158309

Level: 
C
Students of a class take a multiple choice test consisting of \( 10 \) tasks. There are \( 5 \) optional answers to each of the tasks, while only one answer is correct. One student, however, did not study for the test at all. Therefore, he circles his answers randomly, without performing any calculations. Find the probability that he will circle at least \( 3 \) correct answers. Round the result to four decimal places.
\( 0{.}3222 \)
\( 0{.}8591 \)
\( 0{.}1409 \)
\( 0{.}6778 \)

1003158308

Level: 
C
The probability that the randomly selected product is first-class quality is \( 0{.}12 \). Determine the probability that at least \( 2 \) out of \( 50 \) randomly selected products are first-class quality. Round the result to four decimal places.
\( 0{.}9869 \)
\( 0{.}9689 \)
\( 0{.}8969 \)
\( 0{.}8699 \)
\( 0{.}9896 \)
\( 0{.}8996 \)

1003158307

Level: 
C
Suppose that the success rate of one specific medical treatment is \( 90\,\% \). If the treatment is given to \( 20 \) new patients, what is the probability that it is effective in at least \( 18 \) of them? Round the result to four decimal places.
\( 0{.}6769 \)
\( 0{.}9000 \)
\( 0{.}2852 \)
\( 0{.}7148 \)
\( 0{.}8100 \)