Question

Имеются четыре кандидата: Владимир Владимирович, Василий Всеволодович, Вадим Владимирович и Владимир Венедиктович. Из них случайно выбирают двоих. Какова вероятность того, что: а) будет выбран Владимир Венедиктович; б) отца одного из кандидатов зовут так же, как и самого кандидата; в) будут выбраны кандидаты с одинаковыми именами; г) будут выбраны кандидаты с разными отчествами?

Answer 1

А) То, что Владимира Венедиктовича выберут первым из двоих, вероятность 1/4. То, что его первым не выберут - вероятность 3/4.
Если не выбрали, то вероятность, что его выберут вторым, 1/3.
Итого вероятность, что Владимира Венедиктовича выберут:
P = 1/4 + 3/4*1/3 = 1/4 + 1/4 = 2/4 = 1/2.
б) Отца одного из кандидатов зовут также, как его самого - это только Владимир Владимирович. Ситуация такая же, как в пункте а) с Владимиром Венедиктовичем. Вероятность P = 1/2.
в) Кандидаты с одинаковыми именами - это два Владимира.
Вероятность выбрать первым кого-то из Владимиров, равна 1/2.
Если это случилось, то вероятность выбрать второго Владимира 1/3.
Итого P = 1/2*1/3 = 1/6
г) Этот случай самый трудный.
Если первым будет выбран Василий Всеволодович или Владимир Венедиктович, то второй будет по-любому с другим отчеством.
Вероятность этого равна 1/2.
Если же первым будет выбран Владимир Владимирович или Вадим Владимирович, (вероятность 1/2), то второй будет с другим отчеством с вероятностью 2/3 (только 1 из 3 имеет то же отчество Владимирович).
Итого P = 1/2 + 1/2*2/3 = 1/2 + 1/3 = 3/6 + 2/6 = 5/6.

Answer 2

Probability of Vladimir Venediktovich being chosen:

To calculate the probability of Vladimir Venediktovich being chosen, we need to consider that two candidates will be selected randomly out of the four candidates available. Since there are four candidates in total, the probability of Vladimir Venediktovich being chosen is 1/4 or 0.25.

Probability of a candidate's father having the same name as the candidate:

To calculate the probability of a candidate's father having the same name as the candidate, we need to consider that the names of the candidates and their fathers are independent of each other. Therefore, the probability of a candidate's father having the same name as the candidate is the same as the probability of any candidate having the same name as their father. Without further information about the names of the fathers, we cannot determine this probability.

Probability of candidates being chosen with the same names:

To calculate the probability of candidates being chosen with the same names, we need to consider that two candidates will be selected randomly out of the four candidates available. Since there are four candidates in total and two candidates will be chosen, we can calculate this probability by considering the number of ways to choose two candidates with the same name and dividing it by the total number of possible combinations.

Let's consider the candidates with the same names as a group. There are two candidates with the name Vladimir Vladimirovich, one candidate with the name Vasily Vsevolodovich, and one candidate with the name Vadim Vladimirovich.

The number of ways to choose two candidates with the same name is the number of combinations of two candidates from the group of candidates with the same name, which is given by the formula C(2, 2) = 1.

The total number of possible combinations of two candidates out of the four candidates is given by the formula C(4, 2) = 6.

Therefore, the probability of candidates being chosen with the same names is 1/6 or approximately 0.1667.

Probability of candidates being chosen with different patronymics:

To calculate the probability of candidates being chosen with different patronymics, we need to consider that two candidates will be selected randomly out of the four candidates available. Since there are four candidates in total and two candidates will be chosen, we can calculate this probability by considering the number of ways to choose two candidates with different patronymics and dividing it by the total number of possible combinations.

Let's consider the candidates with different patronymics as a group. There are three candidates with different patronymics: Vladimir Vladimirovich, Vasily Vsevolodovich, and Vadim Vladimirovich.

The number of ways to choose two candidates with different patronymics is the number of combinations of one candidate from the group of candidates with different patronymics and one candidate from the group of candidates with the same name but different patronymics, which is given by the formula C(3, 1) * C(1, 1) = 3.

The total number of possible combinations of two candidates out of the four candidates is given by the formula C(4, 2) = 6.

Therefore, the probability of candidates being chosen with different patronymics is 3/6 or 0.5.

Please note that the probabilities provided here assume that the selection process is truly random and that all candidates have an equal chance of being chosen.