Math Problems Vault

Problem 6.1

Suppose the probability of hitting a target is $1/5$ . Ten independent shots are fired. What is the probability of hitting the target at least twice?

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Problem 6.2

Observations have established that in a certain locality, September has on average 12 rainy days. What is the probability that of eight days chosen at random in that month, three days turn out to be rainy?

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Problem 6.3

(a)(b)

Taking the probability of a boy being born to be $1/2$ , find the probability that in a family with 10 children:

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(a)

there are 5 boys and 5 girls;

(b)

the number of boys is not less than 4 and not more than 7.

Problem 6.4

(a)(b)

Among the cocoons of a certain batch, 30 are colored. What is the probability that among 10 cocoons chosen at random from the batch:

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(a)

3 are colored;

(b)

at most 3 are colored?

Problem 6.5

(a)(b)

From an urn containing:

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(a)

$2n$ white and $2n$ black balls;

(b)

$n$ white and $3n$ black balls,

$2n$ balls are drawn at random (with replacement). What is the probability that among them the numbers of white and black balls are equal?

Problem 6.6

Which is more likely: getting at least one 1 when four dice are rolled, or getting at least one double-1 in 24 rolls of two dice?

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Problem 6.7

(a)(b)(c)(d)

Which is more likely, against an equally matched opponent:

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(a)

to win 3 games out of 4, or 5 out of 8;

(b)

to win at least 3 games out of 4, or at least 5 out of 8;

(c)

to win at most $n$ out of $2n$ games, or more than $n$ out of $2n$ games;

(d)

to win at most $n$ out of $2n+1$ games, or more than $n$ out of $2n+1$ games?

Problem 6.8

A certain mathematician carries two matchboxes, each initially containing $N$ matches. Whenever he needs a match, he chooses one of the boxes at random. Find the probability that when the mathematician first pulls out an empty box, the other box contains $r$ matches.

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Problem 6.9

(A generalization of Banach's matchbox problem, see 6.8.) Someone carries two matchboxes A and B, which initially contained $M$ and $N$ matches respectively. Whenever he needs a match, he takes it from box A with probability $p$ or from box B with probability $1-p$ . Find the probability that when the mathematician first pulls out an empty box, the other box contains $r$ matches.

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Problem 6.10

(a)(b)

Two people, independently of one another, toss a coin $n$ times each.

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(a)

Find the probability that they get the same number of heads.

(b)

Prove that the probability that they get the same number of heads coincides with the probability that between them they get a total of $n$ heads.

Problem 6.11

(a)(b)

Using the Bernoulli scheme, prove the combinatorial identities:

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(a)

$2^n = \displaystyle \sum_{i=0}^{n} C_n^i$ ;

(b)

$C_{2n}^{n} = \displaystyle \sum_{i=0}^{n} (C_n^i)^2$ .

Problem 6.12

Find the limit as $n \to \infty$ of the probability that when a fair coin is tossed $n$ times, the number of heads is exactly equal to the number of tails.

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Problem 6.13

The probability of a successful outcome in a complex chemical experiment is $2/3$ . Find the most likely number of successful experiments out of a total of 7.

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Problem 6.14

A battery fired 14 shots at an object, with the probability of a hit on each shot being $1/5$ . Find the most likely number of hits and the probability of that number of hits.

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Problem 6.15

The probability of hitting a target on each shot from a gun is $4/5$ . How many shots must be fired so that the most likely number of hits is 20?

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Problem 6.16

(a)(b)

A fair coin is tossed until the same side comes up twice in a row. Find the probabilities of the following events:

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(a)

the experiment ends before the sixth toss;

(b)

an even number of tosses is required.

Problem 6.17

(a)(b)

A coin is tossed until heads appears for the $m$ -th time. Find the probability that the experiment ends on the $n$ -th toss, if:

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(a)

the coin is fair;

(b)

the coin is biased, i.e., the probability of heads is $p \neq 1/2$ .

Problem 6.18

Two people play a game, taking turns tossing a coin. The winner is whoever is first to get tails. What is the probability of winning for the player who starts the game?

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Problem 6.19

(a)(b)(c)

Quality control inspects items, each of which, independently of the others, may turn out defective with probability $p$ . Find the probability that:

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(a)

out of 10 items inspected, only one turns out defective;

(b)

the first defective item is the $k$ -th one inspected;

(c)

the next 10 items turn out non-defective given that the preceding $l = 5$ items were also non-defective. Does this probability depend on $l$ ?

Problem 6.20

(a)(b)

Six points are thrown at random, independently of one another, onto a segment $AB$ of length $a$ . Find the probability that:

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(a)

two points are at a distance less than $b$ from point $A$ , and four are at a distance greater than $b$ ;

(b)

two points are at a distance less than $c$ from $A$ , one is at a distance not less than $c$ and not greater than $b$ , and three points are at a distance greater than $b$ .

Problem 6.21

What is the probability of getting each face exactly twice when twelve dice are rolled?

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Problem 6.22

Two chess players $A$ and $B$ agree to play a match under the following conditions: $A$ must score 12 points to win (a win counts as one point), $B$ must score 6 points, and draws do not count. $A$ usually wins twice as often as $B$ , counting only decisive games, so the probability of $A$ 's winning may be taken as $2/3$ . The match had to be stopped after $A$ scored 8 points and $B$ scored 4 points. It was decided to award victory to whichever player has the greater probability of eventually winning. Who is the winner?

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Problem 6.23

(a)(b)

In a Bernoulli scheme, the probability of success is $p$ and the probability of failure is $q = 1-p$ . Find the probability that:

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(a)

the pattern FF (two failures in a row) appears before the pattern FS (a failure followed by a success);

(b)

the pattern FF appears before the pattern SF.

Problem 6.24

(a)(b)(c)

Player A simultaneously rolls three dice, while player B at the same time rolls two dice. They repeat these trials in succession until a ``6'' first appears on at least one die. Find the probabilities of the following events:

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(a)

$A = \left\{ \text{a ``6'' first appeared for player A, but not for B}\right\}$ ;

(b)

$B = \left\{ \text{a ``6'' first appeared for player B, but not for A}\right\}$ ;

(c)

$C = \left\{ \text{a ``6'' first appeared simultaneously for A and B}\right\}$ .

Problem 6.25

When a deck of 52 cards is dealt to four players, one of them received no aces three times in a row. Does he have grounds to complain of ``bad luck''?

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Problem 6.26

(a)(b)

An educational institution has 730 students enrolled. A randomly chosen student's birthday falls on a particular day of the year with probability $1/365$ for each of the 365 days. Find:

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(a)

the most likely number of students born on January 1;

(b)

the probability that there exist three students sharing the same birthday.