Probability

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Level:

We roll red and yellow dice. Let the event

A

be: On the red die a number greater than

2

is rolled. And let the event

B

be: The sum of points rolled on both dice is greater than

6

. Find

P (A | B)

. (Note: In the table the sums of points on both dice are listed.)

\frac{3}{4}

\frac{1}{2}

\frac{6}{7}

\frac{1}{4}

1103158401

Level:

We roll red and yellow dice. Find the probability that the yellow die is two given that the sum of both dice is eight. (Note: In the following table, the sums of points on both dice are listed.)

\frac{1}{5}

\frac{1}{6}

\frac{1}{36}

\frac{5}{36}

1003029305

Level:

The production process of a particular component consists of three independent phases. Through long-term monitoring of production quality were found success rates of individual phases

90 %

80 %

and

85 %

. If all three phases of the process are successful, the component is of good quality. What is the probability of producing a good quality component?

0.612

0.003

0.388

0.997

1003029304

Level:

The probability of a man hitting a target is

0.9

. What is the probability that he hits the target twice in a row? Round the result to two decimal places.

0.81

0.99

0.95

0.18

1003029303

Level:

The probability of a man hitting a target is

0.9

. If the man shoots three times, what is the probability that he hits the target at least once? Round the result to three decimal places.

0.999

0.729

0.027

0.243

1003029302

Level:

The quality inspection found that

85 %

of the items are without defect, exactly one defect has

10 %

of the items, and the other items have more than one defect. We pick one item randomly. What is the probability that this item has at least one defect?

0.15

0.10

0.95

0.01

1003029301

Level:

Two regular dice are thrown. Find the probability that the sum is five or six. The results are rounded to two decimal places.

\frac{1}{4} = 0.25

\frac{5}{36} ≐ 0.14

\frac{2}{11} ≐ 0.18

\frac{10}{36} ≐ 0.28

1003029206

Level:

In the hospital,

22

boys and

18

girls were born in one month. Babies were listed in the register by their date of birth. Find the probability that there are at least three boys in the first five places of the register. The results are rounded to four decimal places.

\frac{(\binom{22}{3}) \cdot (\binom{18}{2}) + (\binom{22}{4}) \cdot (\binom{18}{1}) + (\binom{22}{5}) \cdot (\binom{18}{0})}{(\binom{40}{5})} = 0.5982

\frac{(\binom{22}{3}) + (\binom{22}{4}) + (\binom{22}{5})}{(\binom{40}{5})} = 0.0535

\frac{22^{3} \cdot 18^{2} + 22^{4} \cdot 18^{1} + 22^{5} \cdot 18^{0}}{40^{5}} = 0.1252

\frac{(\binom{22}{3}) \cdot (\binom{18}{2}) + (\binom{22}{4}) \cdot (\binom{18}{1}) + (\binom{22}{5}) \cdot (\binom{18}{0})}{40^{5}} = 0.0038

1003029205

Level:

In the hospital,

22

boys and

18

girls were born in one month. Babies were listed in the register by their date of birth. Find the probability that there are two boys and three girls in the first five places of the register. The results are rounded to four decimal places.

\frac{(\binom{22}{2}) \cdot (\binom{18}{3})}{(\binom{40}{5})} = 0.2865

\frac{(\binom{22}{2}) \cdot (\binom{18}{3})}{\frac{40!}{35!}} = 0.0024

\frac{22^{2} \cdot 18^{3}}{40^{5}} = 0.0276

\frac{(\binom{22}{3}) \cdot (\binom{18}{2})}{\frac{40!}{35!}} = 0.0030

1003029204

Level:

A class consists of

50

students including twins, Mark and Martin. For an exam students are randomly divided into two equally sized subgroups. Find the probability that Mark and Martin will be in the same subgroup. The results are rounded to two decimal places.

\frac{(\binom{48}{23}) + (\binom{48}{25})}{(\binom{50}{25})} = 0.49

\frac{(\binom{48}{23})}{(\binom{50}{25})} = 0.24

\frac{2 \cdot (\binom{48}{24})}{(\binom{50}{25})} = 0.51

\frac{(\binom{49}{24})}{(\binom{50}{25})} = 0.50

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