Question

15 БАЛЛОВ! Два действительных числа случайным образом выбираются из отрезка [0,6]. Какова вероятность того, что сумма двух чисел меньше 5, а их произведение меньше 6?​ решать с помощью геометрического определения вероятности

Answer 1

Пусть х - первое число, у - второе число. По условию, $x+y$ откуда $y$ и $xy$ откуда $y$

Найдем точки пересечения графиков функций $y=5-x$ и $y=\dfrac{6}{x}$. Для этого приравниваем функции

$5-x=\dfrac{6}{x}~~~\bigg|\cdot x\\ \\ x^2-5x+6=0$

По теореме Виета

$x_1=2\\ x_2=3$

Смотрим рисунок. Разобьем заштрихованную фигуру прямыми x = 2 и x = 3 и найдем площади $S_{QRBC}=S_1,~ S_{BADC}=S_2,~ S_{ADS}=S_3$

$S_1=\dfrac{QR+BC}{2}\cdot CQ=\dfrac{5+3}{2}\cdot 2=8$ кв. ед.

$\displaystyle S_2=\int\limits^3_2 \dfrac{6}{x}dx=6\ln|x|~\bigg|^3_2=6\Big(\ln 3-\ln 2\Big)=6\ln\dfrac{3}{2}$ кв. ед.

$S_3=\dfrac{AD\cdot DS}{2}=\dfrac{2\cdot 2}{2}=2$ кв. ед.

Площадь заштрихованной фигуры: $S=S_1+S_2+S_3=10+6\ln\dfrac{3}{2}$

Искомая вероятность:  $P=\dfrac{S}{S_{QNOP}}=\dfrac{10+6\ln\dfrac{3}{2}}{6\cdot6}=\dfrac{5+3\ln\dfrac{3}{2}}{18}\approx0.345$

Answer 2

#### Probability of the Sum and Product of Two Randomly Chosen Numbers To find the probability that the sum of two randomly chosen numbers from the interval [0, 6] is less than 5 and their product is less than 6, we can use the geometric definition of probability. The geometric definition of probability states that the probability of an event is equal to the ratio of the favorable outcomes to the total number of possible outcomes. Let's break down the problem step by step: 1. Determine the favorable outcomes: - The sum of two numbers is less than 5: The possible favorable outcomes for the sum are (0, 0), (0, 1), (0, 2), (0, 3), (0, 4), (1, 0), (1, 1), (1, 2), (1, 3), (2, 0), (2, 1), (2, 2), (3, 0), (3, 1), and (4, 0). There are 15 favorable outcomes for the sum. - The product of two numbers is less than 6: The possible favorable outcomes for the product are (0, 0), (0, 1), (0, 2), (0#### Probability of the Sum and Product of Two Randomly Selected Numbers To find the probability that the sum of two randomly selected numbers from the interval [0, 6] is less than 5 and their product is less than 6, we can use the geometric definition of probability. The geometric definition of probability states that the probability of an event is equal to the ratio of the favorable outcomes to the total number of possible outcomes. Let's break down the problem step by step: 1. Determine the favorable outcomes: - The sum of two numbers is less than 5: The possible pairs of numbers that satisfy this condition are (0, 0), (0, 1), (0, 2), (0, 3), (0, 4), (1, 0), (1, 1), (1, 2), (1, 3), (2, 0), (2, 1), (2, 2), (3, 0), and (4, 0). There are 14 favorable outcomes. - The product of two numbers is less than 6: The possible pairs of numbers that satisfy this condition are (0, 0), (0, 1), (0, 2), (0, 3), (0, 4), (0, 5), (1, 0), (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (2, 0), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (3, 0), (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (4, 0), (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (5, 0), (5, 1), (5, 2), (5, 3), (5, 4), and (5, 5). There are 36 favorable outcomes. 2. Determine the total number of possible outcomes: - Since each number can take any value from 0 to 6, there are 7 possible values for each number. Therefore, the total number of possible outcomes is 7 * 7 = 49. 3. Calculate the probability: - The probability is given by the ratio of the favorable outcomes to the total number of possible outcomes: Probability = Favorable outcomes / Total number of outcomes Probability = (14 * 36) / 49 Probability = 504 / 49 Probability ≈ 10.29% Therefore, the probability that the sum of two randomly selected numbers from the interval [0, 6] is less than 5 and their product is less than 6 is approximately **10.29%**. Please note that the above calculation assumes that the selection of numbers is truly random and that each number has an equal chance of being selected.