Question

Помогите решить задание (нужно подробно расписать) 2. Українська співачка дає автограф з імовірністю 0,8, а польська - з імовірністю 0,5. Яка ймовірність того, що завтра після концерту за участю обох зірок: а) вам пощастить одержати обидва автографи; б) пощастить одержати хоча б один автограф; в) не вдасться одержати автограф у польської співачки? 3. Дві дівчини беруть участь у міжнародному конкурсі зі світової економіки. Успішно пройти тур перша дівчина може з імовірністю 0,8, друга - 0,6. Учора пройшов третій, останній тур змагань. Яка імовірність того, що у другої учасниці успішно пройдених турів більше, ніж у першої?

Answer 1

1. Для відповіді на ці питання ми можемо використовувати теорію ймовірностей.

а) Імовірність отримати обидва автографи: 0.8 (Українська співачка) * 0.5 (Польська співачка) = 0.4.

б) Можливість отримати хоча б один автограф: Це одно 1 мінус можливість не отримати жодного автографа. Імовірність не отримати автограф від української співачки: 1 – 0.8 = 0.2. Імовірність не отримати автограф від польської співачки: 1 – 0.5 = 0.5. Тепер ми можемо помножити можливість не отримати автограф від обох співачок: 0.2 * 0.5 = 0.1. Нарешті, можливість отримати хоча б один автограф: 1 - 0.1 = 0.9.

в) Імовірність не отримати автограф у польської співачки: 1 – 0.5 = 0.5.

2. Для цього питання, давайте розглянемо всі можливі варіанти, коли у першої учасниці успішно пройдених турів більше, ніж у другої:

- Перша пройшла 3 тури, друга 2 тури: Імовірність цієї події = 0.8^3 * 0.4^2.

- Перша пройшла 3 тури, друга 1 тур: Імовірність цієї події = 0.8^3*0.4.

- Перша пройшла 2 тури, друга 1 тур: Імовірність цієї події = 0.8^2*0.2.

Тепер складемо ймовірності всіх цих подій, щоб знайти загальну ймовірність, що у другої учасниці успішно пройдених турів більше, ніж у першої.

Answer 2

Probability of obtaining autographs from Ukrainian and Polish singers

To calculate the probability of obtaining autographs from both the Ukrainian and Polish singers, as well as the probability of obtaining at least one autograph, we can use the concept of independent events.

Let's denote the probability of obtaining an autograph from the Ukrainian singer as P(U), which is 0.8, and the probability of obtaining an autograph from the Polish singer as P(P), which is 0.5.

a) Probability of obtaining both autographs: To calculate the probability of obtaining both autographs, we need to multiply the individual probabilities of obtaining each autograph. This is because the events are independent, meaning that the outcome of one event does not affect the outcome of the other event.

So, the probability of obtaining both autographs is given by: P(U and P) = P(U) * P(P) = 0.8 * 0.5 = 0.4 [[1]].

b) Probability of obtaining at least one autograph: To calculate the probability of obtaining at least one autograph, we can use the complement rule. The complement of obtaining at least one autograph is the event of not obtaining any autograph.

So, the probability of not obtaining any autograph is given by: P(not obtaining any autograph) = 1 - P(obtaining at least one autograph).

To calculate the probability of obtaining at least one autograph, we can use the complement of not obtaining any autograph: P(obtaining at least one autograph) = 1 - P(not obtaining any autograph).

The probability of not obtaining any autograph is the product of the probabilities of not obtaining each autograph: P(not obtaining any autograph) = (1 - P(U)) * (1 - P(P)) = (1 - 0.8) * (1 - 0.5) = 0.2 * 0.5 = 0.1 [[2]].

Therefore, the probability of obtaining at least one autograph is: P(obtaining at least one autograph) = 1 - P(not obtaining any autograph) = 1 - 0.1 = 0.9.

c) Probability of not obtaining an autograph from the Polish singer: The probability of not obtaining an autograph from the Polish singer is simply the complement of obtaining an autograph from the Polish singer: P(not obtaining autograph from Polish singer) = 1 - P(P) = 1 - 0.5 = 0.5.

Probability of the second participant having more successful rounds

To calculate the probability of the second participant having more successful rounds than the first participant, we need to consider the probabilities of success for each participant.

Let's denote the probability of the first participant having a successful round as P1 (0.8) and the probability of the second participant having a successful round as P2 (0.6).

To calculate the probability of the second participant having more successful rounds, we need to consider all possible outcomes where the second participant has more successful rounds than the first participant.

There are three possible outcomes: 1. The first participant has 0 successful rounds, and the second participant has 1 or more successful rounds. 2. The first participant has 1 successful round, and the second participant has 2 or more successful rounds. 3. The first participant has 2 successful rounds, and the second participant has 3 successful rounds.

The probability of each outcome can be calculated by multiplying the probabilities of success for each participant.

1. The probability of the first participant having 0 successful rounds and the second participant having 1 or more successful rounds is: P(0 rounds for P1 and 1 or more rounds for P2) = (1 - P1) * P2 = (1 - 0.8) * 0.6 = 0.2 * 0.6 = 0.12.

2. The probability of the first participant having 1 successful round and the second participant having 2 or more successful rounds is: P(1 round for P1 and 2 or more rounds for P2) = P1 * (1 - P2) * P2 = 0.8 * (1 - 0.6) * 0.6 = 0.8 * 0.4 * 0.6 = 0.192.

3. The probability of the first participant having 2 successful rounds and the second participant having 3 successful rounds is: P(2 rounds for P1 and 3 rounds for P2) = P1 * P2 * P2 = 0.8 * 0.6 * 0.6 = 0.288.

To calculate the overall probability of the second participant having more successful rounds, we sum up the probabilities of all three outcomes: P(second participant has more successful rounds) = P(0 rounds for P1 and 1 or more rounds for P2) + P(1 round for P1 and 2 or more rounds for P2) + P(2 rounds for P1 and 3 rounds for P2) = 0.12 + 0.192 + 0.288 = 0.6.

Therefore, the probability of the second participant having more successful rounds than the first participant is 0.6.

Note: The probabilities mentioned in this response are calculated based on the given information and assumptions.