Question

На четырех карточках написаны цифры 1,2,3 и 4 .Выбирают наугад две карточки и кладут их рядом,получив двузначное число.Найдите вероятность следующего события: a) A - получили число с разными цифрами б) B - получили нечётное число в) C - получили число, сумма цифр которого равна 5

Answer 1

Готовое решение на фото.

Answer 2

Problem Analysis

We have four cards labeled with the numbers 1, 2, 3, and 4. We randomly select two cards and place them side by side to form a two-digit number. We need to find the probability of the following events: a) A: The two-digit number has different digits. b) B: The two-digit number is odd. c) C: The sum of the digits in the two-digit number is 5.

Solution

To solve this problem, we need to determine the total number of possible outcomes and the number of favorable outcomes for each event.

Total Number of Possible Outcomes

The total number of possible outcomes is the number of ways we can select two cards from four without replacement. This can be calculated using the combination formula:

Total Outcomes = C(4, 2) = 4! / (2! * (4-2)!) = 6.

Event A: The two-digit number has different digits.

To calculate the number of favorable outcomes for event A, we need to consider the following possibilities: - Selecting 1 and 2: There are two ways to arrange these digits (12 and 21). - Selecting 1 and 3: There are two ways to arrange these digits (13 and 31). - Selecting 1 and 4: There are two ways to arrange these digits (14 and 41). - Selecting 2 and 3: There are two ways to arrange these digits (23 and 32). - Selecting 2 and 4: There are two ways to arrange these digits (24 and 42). - Selecting 3 and 4: There are two ways to arrange these digits (34 and 43).

Therefore, the number of favorable outcomes for event A is 2 + 2 + 2 + 2 + 2 + 2 = 12.

Event B: The two-digit number is odd.

To calculate the number of favorable outcomes for event B, we need to consider the following possibilities: - Selecting 1 and 3: There are two ways to arrange these digits (13 and 31). - Selecting 1 and 4: There are two ways to arrange these digits (14 and 41). - Selecting 3 and 1: There are two ways to arrange these digits (31 and 13). - Selecting 3 and 4: There are two ways to arrange these digits (34 and 43). - Selecting 4 and 1: There are two ways to arrange these digits (41 and 14). - Selecting 4 and 3: There are two ways to arrange these digits (43 and 34).

Therefore, the number of favorable outcomes for event B is 2 + 2 + 2 + 2 + 2 + 2 = 12.

Event C: The sum of the digits in the two-digit number is 5.

To calculate the number of favorable outcomes for event C, we need to consider the following possibilities: - Selecting 1 and 4: There are two ways to arrange these digits (14 and 41). - Selecting 2 and 3: There are two ways to arrange these digits (23 and 32). - Selecting 3 and 2: There are two ways to arrange these digits (32 and 23). - Selecting 4 and 1: There are two ways to arrange these digits (41 and 14).

Therefore, the number of favorable outcomes for event C is 2 + 2 + 2 + 2 = 8.

Calculating the Probabilities

To calculate the probabilities of each event, we divide the number of favorable outcomes by the total number of possible outcomes.

a) Probability of event A: P(A) = Number of favorable outcomes for A / Total number of possible outcomes = 12 / 6 = 2. b) Probability of event B: P(B) = Number of favorable outcomes for B / Total number of possible outcomes = 12 / 6 = 2. c) Probability of event C: P(C) = Number of favorable outcomes for C / Total number of possible outcomes = 8 / 6 = 4/3.

Answer

a) The probability of obtaining a two-digit number with different digits is 2/6. b) The probability of obtaining an odd two-digit number is 2/6. c) The probability of obtaining a two-digit number with a digit sum of 5 is 4/3.

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