Probability

1. There are 52 cards in a deck of cards, of which 4 are aces. Anna and Benjamin (father and his daughter) play a game in which each one of them, in turn, draws four cards without return. Whoever draws more aces wins the game. At the beginning of the game rolls a fair die to decide who will start drawing cards. If the result is 5 or 6, Benjamin starts, otherwise Anna starts.
The order of operations: In the first step the dice is rolled and then the person who starts is determined by the outcome . That person draws 4 cards and keeps them. Afterward, the second person draw 4 cards and then they check who got more aces. Then after the round ended all cards returned to the deck.
a.Assuming that a tie cannot occur (if there is a tie, return all the cards to the pack and play another round
 until a decision is made), what is Anna's probability of winning?
b.If one round is played, what is Anna's probability of winning this round? (here equality is considered as no victory)

Benjamin is waiting for the cable technician, so he left Anna to play alone. Anna plays a version of the game
in which she draws five cards without returning. If she draws at least three cards that are not aces, she wins.
c.What is the probability that Anna wins in a single round?
d.Benjamin's technician is delayed, so Anna continues to play more rounds of the game. She decides to play until she wins exactly five rounds. What is the probability that Anna played exactly 25 rounds?

2. There are 30 coins in the bag, 15 gold and 15 silver. Pulling a coin out of the bag. If it is gold, you return it to the bag and take out two more coins from it (with return). If it is silver, you take out two coins from another bag that contains 25 gold coins and 5 silver coins (with return). In both cases, you return the coin taken out in the previous draw into the bag before the next coin is taken out.
a.What is the probability that three gold coins will be drawn?
b.If it is known that the first coin was gold, what is the probability that both the second and third coins are gold?
c.A player was distracted and did not see the first coin, but he saw that the other two coins were gold.
What is the probability that the first coin was also gold?
d.For the events:
A - The second coin issued is gold
B - The third coin issued is gold
Are events A and B independent? Prove your answer by calculation and also explain why
The obtained result (dependent or independent) is logical considering the process.

3.At the martial arts conference n students learn how to break a meter-long concrete brick with only their hands
only once. There is a 0.5 chance that a student will succeed in breaking the brick. Students who succeed
to break the concrete brick it is divided into two parts where the length of the left part is defined according to
The following density function:

f(x)=cx^2 when 0<=x<=1 and 0 otherwise
For those who failed to break the brick, the length of the left part is defined as a meter.
a."Gold breakage" is a symmetrical breakage in which the parts break into (almost) equal parts. Gold breakage
is any situation in which the gap between the short part and the long part of the brick is at most 5 centimeters.
For a student who managed to break the brick, what is the probability of "breaking gold"?
b.A random student is chosen, what is the probability that the length of the right part will be at most 45 centimeters?
c.What is the probability that exactly a quarter of the students will achieve a right side length of at most 45 centimeters?

4. People come to the lecture according to the Poisson distribution with parameter λ (assume independence between the lectures). In the class there are 250 seats, and students are seated in the order of their arrival. Students who arrive after the class is full go home. The lecturer starts the lecture only when the hall is full.The time period allowed to fill the class is one hour.
In all sections of the question, the answer can be left as a sum as long as it is not infinite.
a.What is the probability that the lecture will start?
During a year there are 28 lectures in total.
b.We will define the random variable X as the number of lectures up to the first lecture that does not start, including the lecture that did not start. How is X distributed if it is known that the lecture does not start at least once?
(For example: if the 3rd lecture has not started but the first 2 have, x=3)
Guidance:
• Write the support of X.
• Mark A - the event that at least one lecture did not start during the year. Now try to understand how
take this event into account during the calculation.