Tasks for independent work

RANDOM EVENTS AND THEIR PROBABILITIES

1.4. Tasks for independent work

42

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10. Let

^ `

An , n 1,2,... be a sequence of events. B_m denotes an event, which means that among events A₁,A₂,... the event A_m will occur first of all.

а) Express B_m through A₁,A₂,...,A_m; b) Prove that events ʑଵ,ʑଶ,… are pairwise disjoint;

c) Express ^f

1

n Bnthrough A₁,A₂,....

1.4.2. Finding probabilities by the classical definition of probability

1. The numbers 1,2,...,n are arranged in a random order. Find the probability that the numbers а) 1 and 2;

b) 1, 2 and 3 are placed side by side in the indicate order.

2. Which event is more likely: when throwing six dice at least once, a «unit» (event A) has dropped out; when throwing twelve dice, at least two times a «unit» (event B) falls out?

3. (The task of the player de Mere) Which event is more likely:

at four dice tosses, at least once a «six points» (event A) will fall out;

at twenty-four tosses of two dice at least once the two «sixes» (an event B) will fall out?

4. а) Find the probability that of the three selected digits 2, 1, 0 will be repeated.

b) Solve the same problem for the four selected digits.

Remark. Here and further we will suppose random numbers taken from the general population numbers 0, 1, 2, ..., 9.

5. Find the probability p_r that if you randomly select r digits from the table of random numbers, there will not be a single repetition. Using the Stirling formula, find the approximate value of p₁₀.

6. n balls are randomly distributed into n boxes. Find the probabilities of events:

а) exactly one box will be empty;

b) there will not be of empty boxes in the cases of Maxwell-Boltzmann and Bose-Einstein statistics.

7.The number a is randomly selected from the set {1,2,...,N}.

Denote by P_N the probability that the number a²1 is a multiple of 10.

Find the limit of this probability _N

N P

f

limo .

8. Two numbers [₁ and [₂ are chosen from the set

^

^1,2,...,N

`

consistently, without repla- cement.

а) Find the probability that the second number will be greater than the first number, i.e.

probability P

^

[₂ ![₁

`

.

b) If three numbers were chosen from the given set by the selection scheme without replacement, then what is the probability that the third number will be a number lying between the first two numbers?

9. There are n places in the hall for the spectators. All places in the room are numbered and all tickets are sold.

If the spectator chooses places randomly, then what is the limit when no f of the probability that no spectator will sit on his (indicated in the ticket) place?

10. Dice is tossed n times. Find the probabilities of the following events:

а) All n times the same number of points fall out;

b) At least once the «six» occurs;

c) «Six» occurs exactly one time;

d) «Six» occurs exactly two times.

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11. There are balls numbered from 1 to n in the urn. If the k balls are randomly selected from the urn under the selection scheme without replacement, what is the probability that the numbers of the selected balls (in order of extraction) form an increasing sequence?

12. Nine passengers randomly seated in three cars.

Find the probabilities of the following events:

a) A = {there are 3 passengers in each wagon};

b) B = {there are 4 passengers in one wagon, 3 – in the other and 2 in the third}.

13. r distinguishable balls are arranged into n boxes.

Find the probability that the boxes Nos. 1, 2, ..., n will contain r₁,r₂,...,r_n balls respectively (

1 2 ... _n , _i 0, 1,2,..., r r r r r t i n).

14. There are 2n cards with the numbers of 1 up to 2n and 2n envelopes, on which the same numbers are written. Cards are randomly placed in envelopes (by one card in each envelope). Find the probability that the sum of the numbers on any envelope and the card lying in it is even.

15. n distinguishable balls are placed into n boxes so that it is equally likely for every ball to be in any box.

Find the probabilities of the following events:

а) All the balls are in the box No.1;

b) Ball No.m is in the box No. l;

c) Balls with the numbers i₁,i₂,...,i_k

ⁱj zⁱm, ^jz^m

are in the boxes with the numbers jk

j

j₁, ₂,...,

j_l z j_m,lzm

(respectively).

16. Continuation. Suppose that under the conditions of the preceding problem there are n balls and N boxes.

How will the corresponding probabilities change?

17. Find the probability that when you place r balls in n boxes, exactly m boxes will remain empty (assume the balls are indistinguishable and all distinguishable locations are equally likely).

18. n persons sit down in a row in a random order.

What is the probability that two identified individuals will be nearby?

Find the corresponding probability if n persons sit down at a round table.

19. Continuation. n persons sit down in a row or at a round table in a random order.

Find in this and the other case the probability that between two definite persons there will be exactly r people.

20. By the condition of the game, three dice are thrown and if the sum of the points dropped does not exceed 10, then the first player will win.

Find the probability of winning the first player.

21. n keys are given to a person, only one of them comes to his door.

He tests them consistently (choice without return). It is clear that this process may require 1, 2, ..., n tests.

Show that each of these outcomes has a probability 1/ n.

22. From an urn containing n balls, all the balls are extracted successively by the selection scheme with the return.

What is the probability that all balls were extracted?

23. Testing the statistical hypothesis. A university professor was fined twelve times for illegal night car parking. All twelve penalties were imposed on Tuesday or Thursday.

Find the probability of this event (Did it make sense to rent a garage only on Tuesdays and Thursdays?)

24. Continuation. Of the twelve fines, no fine was imposed on Sunday. Does this testify that fines are not imposed on Sunday?

25. Each of the n identical sticks randomly breaks into two parts – long and short. Then 2n fragments are combined into n pairs, each of which forms a new «stick».

Find the probabilities that in doing so:

а) parts will be connected in the original order;

b) all long parts will be connected by short ones.

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§ 2. Some classical models and distributions 2.1. The Bernoulli scheme. Binomial distribution

Let some experiment be repeated n times and as a result of each experiment an event A may occur or not occur (for example, each experiment is throwing a coin, and an event A is dropping the «tail»). If an event A occurred as a result of the experiment, then we will say that there was a «success», if the event A did not occur, then we will say that there was a «failure». If we denote the result of the i^th experiment by Z_i and write down Z_i = 1, if the «success» was in the i^th experiment, and Z_i = 0, if the

«failure» was in i^th experiment, then in the space of elementary events, corresponding to the n-fold repetition of the original experiment, it can be described as follows:

^

Z Z: Z Z1, 2,...,Z Z_n : _i 0,1 .

`

:

Then let’s consider two positive numbersp q, such that p q 1, and define the probability P Z of an elementary event Z: by the formula

Z, Z pZqⁿ

P (1) where Z Z ₁ ... Z_n is the number of successes.

First of all, let us show the correctness of the definition (1), i.е. implementation of equality ( )P P( ) 1

Z

Z

:

:

¦

^.

Really,

0 : 0

0

1.

n n

n k n k k n k

k

k k k

n k k n k n

n k

P P p q p q A p q

C p q p q

Z Z

Z Z Z Z

Z

: :

:

¦ ¦ ¦ ¦ ¦

¦

(Above we took into account, that for Ak

^

Z:^:Z k

`

the number of its elements is A_k C_n^k ).

If now for any event Aࣛ

^

^{A A: Ž :}

`

we assume, by definition (§1, formula

(1**)) ( ) ( )

A

P A P

Z

Z

¦

 , then we obtain a finite probability space ሺ^:ǡ ࣛǡ ܲሻ (see §1).

If n 1, then the sample space consists only of two points Z₁ 1 («success») and Z₂ 0(«failure»): :

^ `

^0,1 . Naturally, in this case the probability P(1) p is reasonably called the probability of success, and the probability P(0) q 1 p is the probability of failure.

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The sequence of tests (experiments) described above, in which the probability of success is defined by formula (1), is called the Bernoulli scheme (or the Bernoulli series of independent trials – we will explain later why it is so called).

For the Bernoulli scheme with the probability of success p, the probability of event

Ak={exactly k successes occur in n Bernoulli trials}

is equal to

( )

k k

k n k k n k k k n k

n k k n

A A

P k P A P p q A p q C p q

Z Z

Z

 

¦ ¦

^.

The fact that these numbers are indeed the probabilities follows from the relation

0 0

1

n n

k k n k

k n

k k

P A C p q

¦ ¦

^.

A set of probabilities

^

P A P A P A( ), ( ), ( ),..., ( )0 1 2 P A_n

`

is called a binomial distribu- tion (binomial distribution of the number of successes in the sample of size n).

This distribution arises in a wide variety of probability models and plays an extremely important role in the probability theory. To understand the nature of this distribution for n = 5 («symmetrical case») and n 5, n 10, we give the form of the graph of this distribution (see Fig. 1).

Fig. 1. The graph of binomial probability P k_n( ) when p 1 2 and n 5, 10.

In connection with Fig. 1 we pay attention to the following: the probability P k₅( )

takes the greatest value at two points k₁^* 2 and k₂^* 3; but the probabilityP k₁₀( ) takes the greatest value only at one point k^* 5. To understand this difference, let's consider the probabilities

k n k k n k

n k P A P n k C p q

P( ) ( ) ( , )

when k changes.

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We can write

1 1

( , 1) ! !( )! ( )

( , ) ( 1)!( 1)! ! ( 1)

k n k

k n k

P n k n p q k n k p n k

P n k k n k n p q q k

˜

.

Further, since the inequality P n k( , !1) P n k( , ) is equivalent to the inequality (n1)p k! 1, then, if (n1)p k! 1 (or np q k ! ) then the probability P n k( , ) increases with the transition from k to k1; conversely, if (n1)p k 1 (or

np q k), then P n k( , ) decreases with the transition from k to k1. If (n1)p k 1 (or np q k ), then P n k( , 1) P n k( , ).

Definition. The value of k, at which the probability P n k( , ), as a function of k, takes the greatest value

, 0max ( , )

P n k k nP n k

d d is called the most likely number of successes.

From the definition and the above arguments we obtain the following statement:

If (n1)p is not an integer, then ^{k ª} _¬

^{n 1}

^pº_¼^{, where [a}] is the integer part of number a;

If (n1)p is an integer, then there are two most likely numbers of success:

( 1) 1

k n p and k (n1)p.

In conclusion, we note that for the sequence of n independent Bernoulli trials with the probability of success p the probabilities of events

a) not once there was a success, b) at least once there was a success,

c) there were at least k₁ and at most k₂ successes can be found by the follo- wing formulas (prove!):

а) P n( ,0) qⁿ (1 p) ,ⁿ b)

1

( , ) 1 ( ,0) 1 1 (1 )

n n n

k

P n k P n q p

¦

, (2)

c) ^{2 2}

1 1

( , )

k k

k k n k

n

k k k k

P n k C p q

¦ ¦

^.