Solution to problem 9.7.11 from the collection of Kepe O.E.

Problem 9.7.11 from the collection of Kepe O.?. refers to the section "Probability Theory" and is formulated as follows:

“What is the probability that in a group of 20 students that at least three have the same birthday?”

To solve this problem, you need to use the formula for the probability of coincidence of birthdays, which has the form:

P(A) = 1 - P(A'),

where P(A') is the probability that all 20 students have different birthdays.

To find P(A'), you can use the formula for the product of probabilities:

P(A') = (365/365) * (364/365) * (363/365) * ... * (347/365),

where the numerator of each fractional factor corresponds to the number of days in the year, and the denominator corresponds to the number of days in the year, minus the number of the current birthday.

By substituting the values ​​in the formulas and making calculations, you can get the answer to the problem: the probability that in a group of 20 students at least three have the same birthday is approximately 0.41 or 41%.

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Problem 9.7.11 from the collection of problems by Kepe O.?. in mathematics is as follows: given a sequence of numbers a1, a2, ..., an, each of which can be either 1 or -1. It is necessary to find a subsequence of this sequence whose sum of elements is maximum. The answer is this maximum amount.

To solve the problem, you can use dynamic programming. To do this, you can enter an array dp, where dp[i] is the maximum sum of the subsequence ending at element ai. Initially, all elements of dp are equal to zero, except dp[1], which is equal to a1.

Then, for each i from 2 to n, we need to calculate dp[i] as follows: if dp[i-1] is greater than zero, then dp[i] is equal to dp[i-1] + ai, otherwise dp[i] is equal to ai. The maximum sum of the subsequence will be equal to the maximum element in the dp array.

Thus, the solution to problem 9.7.11 from the collection of Kepe O.?. comes down to solving a dynamic programming problem.

Solution to problem 9.7.11 from the collection of Kepe O.?. consists in determining the angular acceleration of a rod that moves in the plane of the drawing. To do this, it is necessary to know the accelerations of points A and B of the rod at some point in time. From the problem conditions it is known that the acceleration of point A is 2 m/s2, and the acceleration of point B is 6 m/s2.

To solve the problem, you can use the formula for determining angular acceleration:

ω = а / r,

where ω is the angular acceleration, and is the linear acceleration, r is the radius of the circle along which the body moves.

The radius of the circle along which the rod moves is equal to half the length of the rod:

r = AB / 2 = 40 / 2 = 20 cm = 0.2 m.

The linear acceleration of point A is 2 m/s2, and the acceleration of point B is 6 m/s2. The average linear acceleration of the rod can be defined as the arithmetic mean between the linear accelerations of points A and B:

a = (aA + aB) / 2 = (2 + 6) / 2 = 4 m/s2.

Now you can determine the angular acceleration of the rod:

ω = a / r = 4 / 0.2 = 20 rad/s2.

Answer: 10.

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