Problems

Anna, Boris and Fred decided to go to a children’s Christmas party. They agreed to meet at the bus stop, but they do not know who will come to what time. Each of them can come at a random time from 15:00 to 16:00. Fred is the most patient of them all: if he comes and finds that neither Anna nor Boris are at the bus stop, then Fred will wait for one of them for 15 minutes, and if he waits for more than 15 minutes and no one arrives he will go to the Christmas party by himself. Boris is less patient: he will only wait for 10 minutes. Anna is very impatient: she will not wait at all. However, if Boris and Fred meet, they will wait for Anna until 16:00. What is the probability that all of them will go to the Christmas party?

Three tired cowboys went into a bar, and hung their hats on the buffalo horn at the entrance. When the cowboys left at night, they were unable to distinguish one hat from another and therefore took the three hats at random. Find the likelihood that none of them took their own hat.

A die is thrown over and over again. Let $P_n$ denote the probability that, at some point, the sum of the points, taken from the numbers that came out on the top face of the die, from all the rolls made, is $n$. Prove that for $n\ge 7$ the equality $P_n=1/6(P_{n-1}+P_{n-2}+\cdots +P_{n-6})$ is true.

On a roulette, any number from 0 to 2007 can be determined with the same probability. The roulette is spun time after time. Let $P_k$ denote the probability that at some point the sum of the numbers that are determined by a ball being thrown on the roulette is $k$. Which number is larger: $P_{2007}$ or $P_{2008}$?

The figure shows the scheme of a go-karting route. The start and finish are at point $A$, and the driver can go along the route as many times as he wants by going to point $A$ and then back onto the circle.

It takes Fred one minute to get from $A$ to $B$ or from $B$ to $A$. It also takes one minute for Fred to go around the ring and he can travel along the ring in an anti-clockwise direction (the arrows in the image indicate the possible direction of movement). Fred does not turn back halfway along the route nor does not stop. He is allowed to be on the track for 10 minutes. Find the number of possible different routes (i.e. sequences of possible routes).

On the occasion of the beginning of the winter holidays all of the boys from class 8B went to the shooting range. It is known that there are $n$ boys in 8B. There are $n$ targets at the shooting range which the class attended. Each of the boys randomly chooses a target, while some of the boys could choose the same target. After this, all of the boys simultaneously attempt to shoot their target. It is known that each of the boys hits their target. The target is considered to be affected if at least one boy has hit it.

a) Find the average number of affected targets.

b) Can the average number of affected targets be less than $n/2$?

$A$ and $B$ shoot in a shooting gallery, but they only have one six-shot revolver with one cartridge. Therefore, they agreed in turn to randomly rotate the drum and shoot. $A$ goes first. Find the probability that the shot will occur when $A$ has the revolver.

Kate and Gina agreed to meet at the underground in the first hour of the afternoon. Kate comes to the meeting place between noon and one o’clock in the afternoon, waits for 10 minutes and then leaves. Gina does the same.

a) What is the probability that they will meet?

b) How will the probability of a meeting change if Gina decides to come earlier than half past twelve, and Kate still decides to come between noon and one o’clock?

c) How will the probability of a meeting change if Gina decides to come at an arbitrary time between 12:00 and 12:50, and Kate still comes between 12:00 and 13:00?

The probability that a purchased lightbulb will work is 0.95. How many light bulbs should I buy so that, with a probability of 0.99, there would be at least 5 that work among them?

A hunter has two dogs. Once, when he was lost in the woods, he went to the fork in the road. The hunter knows that each of the dogs with probability $p$ will choose the way home. He decided to release the dogs in turn. If both choose the same road, he will follow them; if they are separated, the hunter will choose the road, by throwing a coin. Will this increase the hunter’s chances of choosing the way home, compared to if he had only one dog?